01. Vectors 20130305 - Yonsei Universityweb.yonsei.ac.kr/hgjung/Lectures/MAT203/01 Vectors.pdf ·...
Transcript of 01. Vectors 20130305 - Yonsei Universityweb.yonsei.ac.kr/hgjung/Lectures/MAT203/01 Vectors.pdf ·...
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1 Vectors1 Vectors
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11 Vectors and Matrices11 Vectors and Matrices
Linear algebra is concerned with two basic kinds of quantities ldquovectorsrdquo and ldquomatricesrdquo
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
- Scalar a numerical value
denoted by lowercase italic type such as a k v w and x
ex) temperature length and speed
- Vector a numerical value and a direction
denoted by lowercase boldface type such as a k v w and x
ex) velocity force and displacement
Scalars and Vectors
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11 Vectors and Matrices11 Vectors and Matrices
Initial point
Terminal point
MagnitudeDirection
Scalars and Vectors
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11 Vectors and Matrices11 Vectors and Matrices
Bound vector Free vector
Equivalent Vectors
In this text we will focus exclusively on free vectors leaving the study of bound vectors for courses in engineering and physics
Two vectors v and w are equal (also called equivalent) if they are represented by parallel arrows with the same length and direction v=w
The vector whose initial and terminal points coincide has length zero so we call this zero vector and denote it by 0
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11 Vectors and Matrices11 Vectors and Matrices
Vector Addition
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11 Vectors and Matrices11 Vectors and Matrices
Vector Addition
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11 Vectors and Matrices11 Vectors and Matrices
Vector Subtraction
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11 Vectors and Matrices11 Vectors and Matrices
Scalar Multiplication
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11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
Rectangular coordinate system in 2-space
x-axis
y-axis
origin
One-to-one correspondence between points in the plane and ordered pairs of real numbers
P a b
point coordinates P a b
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
Rectangular coordinate system in 3-space
x-axis y-axis z-axis
Left-handed Right-handed
In this text we will work exclusively with right-handed coordinate systems
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
If a vector v in 2-space or 3-space is positioned with its initial point at the origin of a rectangular coordinate system then the vector is completely determined by the coordinates of its terminal point and we call these coordinates the components of v relative to the coordinate system
The set of all vectors in 2-space R2
The set of all vectors in 3-space R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Components of a Vector Whose Initial Point Is Not At The Origin
v is a vector in R2 with initial point P1(x1y1) and terminal point P2(x2y2)
1 2 2 1 2 1 2 1PP OP OP x x y y v
The components of v are obtained by subtracting the coordinates of the initial point from the corresponding coordinates of the terminal point
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Rn
You can think of the numbers in an n-tuple (v1 v2 hellip vn) as either the coordinates of a generalized point or the components of a generalized vector
0=(000hellip0)
We will call this the zero vector or sometimes the origin of Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
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11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
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11 Vectors and Matrices11 Vectors and Matrices
Sums of Three or More Vectors
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11 Vectors and Matrices11 Vectors and Matrices
Parallel and Collinear Vectors
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11 Vectors and Matrices11 Vectors and Matrices
Linear Combinations
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Application to Computer Color Models
Colors on computer monitors are commonly based on what is called the RGB color model
Each color vector c in RGB space or the RGB color cube is expressible as a linear combination of the form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Alternative Notations for Vectors
Comma-delimited form
Row-vector form
Column-vector form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Matrices
We define a matrix to be a rectangular array of numbers called the entries of the matrix
You can also think of a matrix as a list of row vectors or column vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
The length of a vector v in R2 and R3 is commonly denoted by the symbol ||v||
From the theorem of Pythagoras
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Unit Vectors
A vector of length 1 is called a unit vector
Normalizing v
Example 2Example 2
Find the unit vector u that has the same direction as v=(22-1)
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12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors
In R2 these vectors are denoted by
In R3 these vectors are denoted by
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
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12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
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12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
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12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
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12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
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12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
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12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
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12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
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12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
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12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
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Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
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Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
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Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
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Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
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Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
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Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
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Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Linear algebra is concerned with two basic kinds of quantities ldquovectorsrdquo and ldquomatricesrdquo
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
- Scalar a numerical value
denoted by lowercase italic type such as a k v w and x
ex) temperature length and speed
- Vector a numerical value and a direction
denoted by lowercase boldface type such as a k v w and x
ex) velocity force and displacement
Scalars and Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Initial point
Terminal point
MagnitudeDirection
Scalars and Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Bound vector Free vector
Equivalent Vectors
In this text we will focus exclusively on free vectors leaving the study of bound vectors for courses in engineering and physics
Two vectors v and w are equal (also called equivalent) if they are represented by parallel arrows with the same length and direction v=w
The vector whose initial and terminal points coincide has length zero so we call this zero vector and denote it by 0
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vector Addition
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vector Addition
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vector Subtraction
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Scalar Multiplication
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
Rectangular coordinate system in 2-space
x-axis
y-axis
origin
One-to-one correspondence between points in the plane and ordered pairs of real numbers
P a b
point coordinates P a b
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
Rectangular coordinate system in 3-space
x-axis y-axis z-axis
Left-handed Right-handed
In this text we will work exclusively with right-handed coordinate systems
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
If a vector v in 2-space or 3-space is positioned with its initial point at the origin of a rectangular coordinate system then the vector is completely determined by the coordinates of its terminal point and we call these coordinates the components of v relative to the coordinate system
The set of all vectors in 2-space R2
The set of all vectors in 3-space R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Components of a Vector Whose Initial Point Is Not At The Origin
v is a vector in R2 with initial point P1(x1y1) and terminal point P2(x2y2)
1 2 2 1 2 1 2 1PP OP OP x x y y v
The components of v are obtained by subtracting the coordinates of the initial point from the corresponding coordinates of the terminal point
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Rn
You can think of the numbers in an n-tuple (v1 v2 hellip vn) as either the coordinates of a generalized point or the components of a generalized vector
0=(000hellip0)
We will call this the zero vector or sometimes the origin of Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
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11 Vectors and Matrices11 Vectors and Matrices
Sums of Three or More Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Parallel and Collinear Vectors
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11 Vectors and Matrices11 Vectors and Matrices
Linear Combinations
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Application to Computer Color Models
Colors on computer monitors are commonly based on what is called the RGB color model
Each color vector c in RGB space or the RGB color cube is expressible as a linear combination of the form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Alternative Notations for Vectors
Comma-delimited form
Row-vector form
Column-vector form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Matrices
We define a matrix to be a rectangular array of numbers called the entries of the matrix
You can also think of a matrix as a list of row vectors or column vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
The length of a vector v in R2 and R3 is commonly denoted by the symbol ||v||
From the theorem of Pythagoras
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Unit Vectors
A vector of length 1 is called a unit vector
Normalizing v
Example 2Example 2
Find the unit vector u that has the same direction as v=(22-1)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors
In R2 these vectors are denoted by
In R3 these vectors are denoted by
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
- Scalar a numerical value
denoted by lowercase italic type such as a k v w and x
ex) temperature length and speed
- Vector a numerical value and a direction
denoted by lowercase boldface type such as a k v w and x
ex) velocity force and displacement
Scalars and Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Initial point
Terminal point
MagnitudeDirection
Scalars and Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Bound vector Free vector
Equivalent Vectors
In this text we will focus exclusively on free vectors leaving the study of bound vectors for courses in engineering and physics
Two vectors v and w are equal (also called equivalent) if they are represented by parallel arrows with the same length and direction v=w
The vector whose initial and terminal points coincide has length zero so we call this zero vector and denote it by 0
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vector Addition
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vector Addition
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vector Subtraction
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Scalar Multiplication
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
Rectangular coordinate system in 2-space
x-axis
y-axis
origin
One-to-one correspondence between points in the plane and ordered pairs of real numbers
P a b
point coordinates P a b
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
Rectangular coordinate system in 3-space
x-axis y-axis z-axis
Left-handed Right-handed
In this text we will work exclusively with right-handed coordinate systems
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
If a vector v in 2-space or 3-space is positioned with its initial point at the origin of a rectangular coordinate system then the vector is completely determined by the coordinates of its terminal point and we call these coordinates the components of v relative to the coordinate system
The set of all vectors in 2-space R2
The set of all vectors in 3-space R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Components of a Vector Whose Initial Point Is Not At The Origin
v is a vector in R2 with initial point P1(x1y1) and terminal point P2(x2y2)
1 2 2 1 2 1 2 1PP OP OP x x y y v
The components of v are obtained by subtracting the coordinates of the initial point from the corresponding coordinates of the terminal point
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Rn
You can think of the numbers in an n-tuple (v1 v2 hellip vn) as either the coordinates of a generalized point or the components of a generalized vector
0=(000hellip0)
We will call this the zero vector or sometimes the origin of Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Sums of Three or More Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Parallel and Collinear Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Linear Combinations
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Application to Computer Color Models
Colors on computer monitors are commonly based on what is called the RGB color model
Each color vector c in RGB space or the RGB color cube is expressible as a linear combination of the form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Alternative Notations for Vectors
Comma-delimited form
Row-vector form
Column-vector form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Matrices
We define a matrix to be a rectangular array of numbers called the entries of the matrix
You can also think of a matrix as a list of row vectors or column vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
The length of a vector v in R2 and R3 is commonly denoted by the symbol ||v||
From the theorem of Pythagoras
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Unit Vectors
A vector of length 1 is called a unit vector
Normalizing v
Example 2Example 2
Find the unit vector u that has the same direction as v=(22-1)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors
In R2 these vectors are denoted by
In R3 these vectors are denoted by
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Initial point
Terminal point
MagnitudeDirection
Scalars and Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Bound vector Free vector
Equivalent Vectors
In this text we will focus exclusively on free vectors leaving the study of bound vectors for courses in engineering and physics
Two vectors v and w are equal (also called equivalent) if they are represented by parallel arrows with the same length and direction v=w
The vector whose initial and terminal points coincide has length zero so we call this zero vector and denote it by 0
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vector Addition
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vector Addition
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vector Subtraction
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Scalar Multiplication
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
Rectangular coordinate system in 2-space
x-axis
y-axis
origin
One-to-one correspondence between points in the plane and ordered pairs of real numbers
P a b
point coordinates P a b
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
Rectangular coordinate system in 3-space
x-axis y-axis z-axis
Left-handed Right-handed
In this text we will work exclusively with right-handed coordinate systems
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
If a vector v in 2-space or 3-space is positioned with its initial point at the origin of a rectangular coordinate system then the vector is completely determined by the coordinates of its terminal point and we call these coordinates the components of v relative to the coordinate system
The set of all vectors in 2-space R2
The set of all vectors in 3-space R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Components of a Vector Whose Initial Point Is Not At The Origin
v is a vector in R2 with initial point P1(x1y1) and terminal point P2(x2y2)
1 2 2 1 2 1 2 1PP OP OP x x y y v
The components of v are obtained by subtracting the coordinates of the initial point from the corresponding coordinates of the terminal point
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Rn
You can think of the numbers in an n-tuple (v1 v2 hellip vn) as either the coordinates of a generalized point or the components of a generalized vector
0=(000hellip0)
We will call this the zero vector or sometimes the origin of Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Sums of Three or More Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Parallel and Collinear Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Linear Combinations
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Application to Computer Color Models
Colors on computer monitors are commonly based on what is called the RGB color model
Each color vector c in RGB space or the RGB color cube is expressible as a linear combination of the form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Alternative Notations for Vectors
Comma-delimited form
Row-vector form
Column-vector form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Matrices
We define a matrix to be a rectangular array of numbers called the entries of the matrix
You can also think of a matrix as a list of row vectors or column vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
The length of a vector v in R2 and R3 is commonly denoted by the symbol ||v||
From the theorem of Pythagoras
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Unit Vectors
A vector of length 1 is called a unit vector
Normalizing v
Example 2Example 2
Find the unit vector u that has the same direction as v=(22-1)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors
In R2 these vectors are denoted by
In R3 these vectors are denoted by
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Bound vector Free vector
Equivalent Vectors
In this text we will focus exclusively on free vectors leaving the study of bound vectors for courses in engineering and physics
Two vectors v and w are equal (also called equivalent) if they are represented by parallel arrows with the same length and direction v=w
The vector whose initial and terminal points coincide has length zero so we call this zero vector and denote it by 0
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vector Addition
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vector Addition
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vector Subtraction
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Scalar Multiplication
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
Rectangular coordinate system in 2-space
x-axis
y-axis
origin
One-to-one correspondence between points in the plane and ordered pairs of real numbers
P a b
point coordinates P a b
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
Rectangular coordinate system in 3-space
x-axis y-axis z-axis
Left-handed Right-handed
In this text we will work exclusively with right-handed coordinate systems
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
If a vector v in 2-space or 3-space is positioned with its initial point at the origin of a rectangular coordinate system then the vector is completely determined by the coordinates of its terminal point and we call these coordinates the components of v relative to the coordinate system
The set of all vectors in 2-space R2
The set of all vectors in 3-space R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Components of a Vector Whose Initial Point Is Not At The Origin
v is a vector in R2 with initial point P1(x1y1) and terminal point P2(x2y2)
1 2 2 1 2 1 2 1PP OP OP x x y y v
The components of v are obtained by subtracting the coordinates of the initial point from the corresponding coordinates of the terminal point
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Rn
You can think of the numbers in an n-tuple (v1 v2 hellip vn) as either the coordinates of a generalized point or the components of a generalized vector
0=(000hellip0)
We will call this the zero vector or sometimes the origin of Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Sums of Three or More Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Parallel and Collinear Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Linear Combinations
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Application to Computer Color Models
Colors on computer monitors are commonly based on what is called the RGB color model
Each color vector c in RGB space or the RGB color cube is expressible as a linear combination of the form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Alternative Notations for Vectors
Comma-delimited form
Row-vector form
Column-vector form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Matrices
We define a matrix to be a rectangular array of numbers called the entries of the matrix
You can also think of a matrix as a list of row vectors or column vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
The length of a vector v in R2 and R3 is commonly denoted by the symbol ||v||
From the theorem of Pythagoras
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Unit Vectors
A vector of length 1 is called a unit vector
Normalizing v
Example 2Example 2
Find the unit vector u that has the same direction as v=(22-1)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors
In R2 these vectors are denoted by
In R3 these vectors are denoted by
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vector Addition
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vector Addition
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vector Subtraction
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Scalar Multiplication
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
Rectangular coordinate system in 2-space
x-axis
y-axis
origin
One-to-one correspondence between points in the plane and ordered pairs of real numbers
P a b
point coordinates P a b
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
Rectangular coordinate system in 3-space
x-axis y-axis z-axis
Left-handed Right-handed
In this text we will work exclusively with right-handed coordinate systems
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
If a vector v in 2-space or 3-space is positioned with its initial point at the origin of a rectangular coordinate system then the vector is completely determined by the coordinates of its terminal point and we call these coordinates the components of v relative to the coordinate system
The set of all vectors in 2-space R2
The set of all vectors in 3-space R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Components of a Vector Whose Initial Point Is Not At The Origin
v is a vector in R2 with initial point P1(x1y1) and terminal point P2(x2y2)
1 2 2 1 2 1 2 1PP OP OP x x y y v
The components of v are obtained by subtracting the coordinates of the initial point from the corresponding coordinates of the terminal point
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Rn
You can think of the numbers in an n-tuple (v1 v2 hellip vn) as either the coordinates of a generalized point or the components of a generalized vector
0=(000hellip0)
We will call this the zero vector or sometimes the origin of Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Sums of Three or More Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Parallel and Collinear Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Linear Combinations
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Application to Computer Color Models
Colors on computer monitors are commonly based on what is called the RGB color model
Each color vector c in RGB space or the RGB color cube is expressible as a linear combination of the form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Alternative Notations for Vectors
Comma-delimited form
Row-vector form
Column-vector form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Matrices
We define a matrix to be a rectangular array of numbers called the entries of the matrix
You can also think of a matrix as a list of row vectors or column vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
The length of a vector v in R2 and R3 is commonly denoted by the symbol ||v||
From the theorem of Pythagoras
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Unit Vectors
A vector of length 1 is called a unit vector
Normalizing v
Example 2Example 2
Find the unit vector u that has the same direction as v=(22-1)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors
In R2 these vectors are denoted by
In R3 these vectors are denoted by
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vector Addition
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vector Subtraction
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Scalar Multiplication
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
Rectangular coordinate system in 2-space
x-axis
y-axis
origin
One-to-one correspondence between points in the plane and ordered pairs of real numbers
P a b
point coordinates P a b
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
Rectangular coordinate system in 3-space
x-axis y-axis z-axis
Left-handed Right-handed
In this text we will work exclusively with right-handed coordinate systems
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
If a vector v in 2-space or 3-space is positioned with its initial point at the origin of a rectangular coordinate system then the vector is completely determined by the coordinates of its terminal point and we call these coordinates the components of v relative to the coordinate system
The set of all vectors in 2-space R2
The set of all vectors in 3-space R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Components of a Vector Whose Initial Point Is Not At The Origin
v is a vector in R2 with initial point P1(x1y1) and terminal point P2(x2y2)
1 2 2 1 2 1 2 1PP OP OP x x y y v
The components of v are obtained by subtracting the coordinates of the initial point from the corresponding coordinates of the terminal point
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Rn
You can think of the numbers in an n-tuple (v1 v2 hellip vn) as either the coordinates of a generalized point or the components of a generalized vector
0=(000hellip0)
We will call this the zero vector or sometimes the origin of Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Sums of Three or More Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Parallel and Collinear Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Linear Combinations
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Application to Computer Color Models
Colors on computer monitors are commonly based on what is called the RGB color model
Each color vector c in RGB space or the RGB color cube is expressible as a linear combination of the form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Alternative Notations for Vectors
Comma-delimited form
Row-vector form
Column-vector form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Matrices
We define a matrix to be a rectangular array of numbers called the entries of the matrix
You can also think of a matrix as a list of row vectors or column vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
The length of a vector v in R2 and R3 is commonly denoted by the symbol ||v||
From the theorem of Pythagoras
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Unit Vectors
A vector of length 1 is called a unit vector
Normalizing v
Example 2Example 2
Find the unit vector u that has the same direction as v=(22-1)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors
In R2 these vectors are denoted by
In R3 these vectors are denoted by
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vector Subtraction
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Scalar Multiplication
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
Rectangular coordinate system in 2-space
x-axis
y-axis
origin
One-to-one correspondence between points in the plane and ordered pairs of real numbers
P a b
point coordinates P a b
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
Rectangular coordinate system in 3-space
x-axis y-axis z-axis
Left-handed Right-handed
In this text we will work exclusively with right-handed coordinate systems
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
If a vector v in 2-space or 3-space is positioned with its initial point at the origin of a rectangular coordinate system then the vector is completely determined by the coordinates of its terminal point and we call these coordinates the components of v relative to the coordinate system
The set of all vectors in 2-space R2
The set of all vectors in 3-space R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Components of a Vector Whose Initial Point Is Not At The Origin
v is a vector in R2 with initial point P1(x1y1) and terminal point P2(x2y2)
1 2 2 1 2 1 2 1PP OP OP x x y y v
The components of v are obtained by subtracting the coordinates of the initial point from the corresponding coordinates of the terminal point
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Rn
You can think of the numbers in an n-tuple (v1 v2 hellip vn) as either the coordinates of a generalized point or the components of a generalized vector
0=(000hellip0)
We will call this the zero vector or sometimes the origin of Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Sums of Three or More Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Parallel and Collinear Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Linear Combinations
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Application to Computer Color Models
Colors on computer monitors are commonly based on what is called the RGB color model
Each color vector c in RGB space or the RGB color cube is expressible as a linear combination of the form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Alternative Notations for Vectors
Comma-delimited form
Row-vector form
Column-vector form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Matrices
We define a matrix to be a rectangular array of numbers called the entries of the matrix
You can also think of a matrix as a list of row vectors or column vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
The length of a vector v in R2 and R3 is commonly denoted by the symbol ||v||
From the theorem of Pythagoras
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Unit Vectors
A vector of length 1 is called a unit vector
Normalizing v
Example 2Example 2
Find the unit vector u that has the same direction as v=(22-1)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors
In R2 these vectors are denoted by
In R3 these vectors are denoted by
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Scalar Multiplication
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
Rectangular coordinate system in 2-space
x-axis
y-axis
origin
One-to-one correspondence between points in the plane and ordered pairs of real numbers
P a b
point coordinates P a b
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
Rectangular coordinate system in 3-space
x-axis y-axis z-axis
Left-handed Right-handed
In this text we will work exclusively with right-handed coordinate systems
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
If a vector v in 2-space or 3-space is positioned with its initial point at the origin of a rectangular coordinate system then the vector is completely determined by the coordinates of its terminal point and we call these coordinates the components of v relative to the coordinate system
The set of all vectors in 2-space R2
The set of all vectors in 3-space R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Components of a Vector Whose Initial Point Is Not At The Origin
v is a vector in R2 with initial point P1(x1y1) and terminal point P2(x2y2)
1 2 2 1 2 1 2 1PP OP OP x x y y v
The components of v are obtained by subtracting the coordinates of the initial point from the corresponding coordinates of the terminal point
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Rn
You can think of the numbers in an n-tuple (v1 v2 hellip vn) as either the coordinates of a generalized point or the components of a generalized vector
0=(000hellip0)
We will call this the zero vector or sometimes the origin of Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Sums of Three or More Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Parallel and Collinear Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Linear Combinations
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Application to Computer Color Models
Colors on computer monitors are commonly based on what is called the RGB color model
Each color vector c in RGB space or the RGB color cube is expressible as a linear combination of the form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Alternative Notations for Vectors
Comma-delimited form
Row-vector form
Column-vector form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Matrices
We define a matrix to be a rectangular array of numbers called the entries of the matrix
You can also think of a matrix as a list of row vectors or column vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
The length of a vector v in R2 and R3 is commonly denoted by the symbol ||v||
From the theorem of Pythagoras
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Unit Vectors
A vector of length 1 is called a unit vector
Normalizing v
Example 2Example 2
Find the unit vector u that has the same direction as v=(22-1)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors
In R2 these vectors are denoted by
In R3 these vectors are denoted by
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
Rectangular coordinate system in 2-space
x-axis
y-axis
origin
One-to-one correspondence between points in the plane and ordered pairs of real numbers
P a b
point coordinates P a b
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
Rectangular coordinate system in 3-space
x-axis y-axis z-axis
Left-handed Right-handed
In this text we will work exclusively with right-handed coordinate systems
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
If a vector v in 2-space or 3-space is positioned with its initial point at the origin of a rectangular coordinate system then the vector is completely determined by the coordinates of its terminal point and we call these coordinates the components of v relative to the coordinate system
The set of all vectors in 2-space R2
The set of all vectors in 3-space R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Components of a Vector Whose Initial Point Is Not At The Origin
v is a vector in R2 with initial point P1(x1y1) and terminal point P2(x2y2)
1 2 2 1 2 1 2 1PP OP OP x x y y v
The components of v are obtained by subtracting the coordinates of the initial point from the corresponding coordinates of the terminal point
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Rn
You can think of the numbers in an n-tuple (v1 v2 hellip vn) as either the coordinates of a generalized point or the components of a generalized vector
0=(000hellip0)
We will call this the zero vector or sometimes the origin of Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Sums of Three or More Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Parallel and Collinear Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Linear Combinations
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Application to Computer Color Models
Colors on computer monitors are commonly based on what is called the RGB color model
Each color vector c in RGB space or the RGB color cube is expressible as a linear combination of the form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Alternative Notations for Vectors
Comma-delimited form
Row-vector form
Column-vector form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Matrices
We define a matrix to be a rectangular array of numbers called the entries of the matrix
You can also think of a matrix as a list of row vectors or column vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
The length of a vector v in R2 and R3 is commonly denoted by the symbol ||v||
From the theorem of Pythagoras
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Unit Vectors
A vector of length 1 is called a unit vector
Normalizing v
Example 2Example 2
Find the unit vector u that has the same direction as v=(22-1)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors
In R2 these vectors are denoted by
In R3 these vectors are denoted by
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
Rectangular coordinate system in 3-space
x-axis y-axis z-axis
Left-handed Right-handed
In this text we will work exclusively with right-handed coordinate systems
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
If a vector v in 2-space or 3-space is positioned with its initial point at the origin of a rectangular coordinate system then the vector is completely determined by the coordinates of its terminal point and we call these coordinates the components of v relative to the coordinate system
The set of all vectors in 2-space R2
The set of all vectors in 3-space R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Components of a Vector Whose Initial Point Is Not At The Origin
v is a vector in R2 with initial point P1(x1y1) and terminal point P2(x2y2)
1 2 2 1 2 1 2 1PP OP OP x x y y v
The components of v are obtained by subtracting the coordinates of the initial point from the corresponding coordinates of the terminal point
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Rn
You can think of the numbers in an n-tuple (v1 v2 hellip vn) as either the coordinates of a generalized point or the components of a generalized vector
0=(000hellip0)
We will call this the zero vector or sometimes the origin of Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Sums of Three or More Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Parallel and Collinear Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Linear Combinations
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Application to Computer Color Models
Colors on computer monitors are commonly based on what is called the RGB color model
Each color vector c in RGB space or the RGB color cube is expressible as a linear combination of the form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Alternative Notations for Vectors
Comma-delimited form
Row-vector form
Column-vector form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Matrices
We define a matrix to be a rectangular array of numbers called the entries of the matrix
You can also think of a matrix as a list of row vectors or column vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
The length of a vector v in R2 and R3 is commonly denoted by the symbol ||v||
From the theorem of Pythagoras
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Unit Vectors
A vector of length 1 is called a unit vector
Normalizing v
Example 2Example 2
Find the unit vector u that has the same direction as v=(22-1)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors
In R2 these vectors are denoted by
In R3 these vectors are denoted by
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Coordinate Systems
If a vector v in 2-space or 3-space is positioned with its initial point at the origin of a rectangular coordinate system then the vector is completely determined by the coordinates of its terminal point and we call these coordinates the components of v relative to the coordinate system
The set of all vectors in 2-space R2
The set of all vectors in 3-space R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Components of a Vector Whose Initial Point Is Not At The Origin
v is a vector in R2 with initial point P1(x1y1) and terminal point P2(x2y2)
1 2 2 1 2 1 2 1PP OP OP x x y y v
The components of v are obtained by subtracting the coordinates of the initial point from the corresponding coordinates of the terminal point
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Rn
You can think of the numbers in an n-tuple (v1 v2 hellip vn) as either the coordinates of a generalized point or the components of a generalized vector
0=(000hellip0)
We will call this the zero vector or sometimes the origin of Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Sums of Three or More Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Parallel and Collinear Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Linear Combinations
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Application to Computer Color Models
Colors on computer monitors are commonly based on what is called the RGB color model
Each color vector c in RGB space or the RGB color cube is expressible as a linear combination of the form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Alternative Notations for Vectors
Comma-delimited form
Row-vector form
Column-vector form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Matrices
We define a matrix to be a rectangular array of numbers called the entries of the matrix
You can also think of a matrix as a list of row vectors or column vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
The length of a vector v in R2 and R3 is commonly denoted by the symbol ||v||
From the theorem of Pythagoras
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Unit Vectors
A vector of length 1 is called a unit vector
Normalizing v
Example 2Example 2
Find the unit vector u that has the same direction as v=(22-1)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors
In R2 these vectors are denoted by
In R3 these vectors are denoted by
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Components of a Vector Whose Initial Point Is Not At The Origin
v is a vector in R2 with initial point P1(x1y1) and terminal point P2(x2y2)
1 2 2 1 2 1 2 1PP OP OP x x y y v
The components of v are obtained by subtracting the coordinates of the initial point from the corresponding coordinates of the terminal point
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Rn
You can think of the numbers in an n-tuple (v1 v2 hellip vn) as either the coordinates of a generalized point or the components of a generalized vector
0=(000hellip0)
We will call this the zero vector or sometimes the origin of Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Sums of Three or More Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Parallel and Collinear Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Linear Combinations
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Application to Computer Color Models
Colors on computer monitors are commonly based on what is called the RGB color model
Each color vector c in RGB space or the RGB color cube is expressible as a linear combination of the form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Alternative Notations for Vectors
Comma-delimited form
Row-vector form
Column-vector form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Matrices
We define a matrix to be a rectangular array of numbers called the entries of the matrix
You can also think of a matrix as a list of row vectors or column vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
The length of a vector v in R2 and R3 is commonly denoted by the symbol ||v||
From the theorem of Pythagoras
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Unit Vectors
A vector of length 1 is called a unit vector
Normalizing v
Example 2Example 2
Find the unit vector u that has the same direction as v=(22-1)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors
In R2 these vectors are denoted by
In R3 these vectors are denoted by
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Vectors in Rn
You can think of the numbers in an n-tuple (v1 v2 hellip vn) as either the coordinates of a generalized point or the components of a generalized vector
0=(000hellip0)
We will call this the zero vector or sometimes the origin of Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Sums of Three or More Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Parallel and Collinear Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Linear Combinations
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Application to Computer Color Models
Colors on computer monitors are commonly based on what is called the RGB color model
Each color vector c in RGB space or the RGB color cube is expressible as a linear combination of the form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Alternative Notations for Vectors
Comma-delimited form
Row-vector form
Column-vector form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Matrices
We define a matrix to be a rectangular array of numbers called the entries of the matrix
You can also think of a matrix as a list of row vectors or column vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
The length of a vector v in R2 and R3 is commonly denoted by the symbol ||v||
From the theorem of Pythagoras
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Unit Vectors
A vector of length 1 is called a unit vector
Normalizing v
Example 2Example 2
Find the unit vector u that has the same direction as v=(22-1)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors
In R2 these vectors are denoted by
In R3 these vectors are denoted by
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Sums of Three or More Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Parallel and Collinear Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Linear Combinations
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Application to Computer Color Models
Colors on computer monitors are commonly based on what is called the RGB color model
Each color vector c in RGB space or the RGB color cube is expressible as a linear combination of the form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Alternative Notations for Vectors
Comma-delimited form
Row-vector form
Column-vector form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Matrices
We define a matrix to be a rectangular array of numbers called the entries of the matrix
You can also think of a matrix as a list of row vectors or column vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
The length of a vector v in R2 and R3 is commonly denoted by the symbol ||v||
From the theorem of Pythagoras
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Unit Vectors
A vector of length 1 is called a unit vector
Normalizing v
Example 2Example 2
Find the unit vector u that has the same direction as v=(22-1)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors
In R2 these vectors are denoted by
In R3 these vectors are denoted by
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Equality of Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Sums of Three or More Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Parallel and Collinear Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Linear Combinations
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Application to Computer Color Models
Colors on computer monitors are commonly based on what is called the RGB color model
Each color vector c in RGB space or the RGB color cube is expressible as a linear combination of the form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Alternative Notations for Vectors
Comma-delimited form
Row-vector form
Column-vector form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Matrices
We define a matrix to be a rectangular array of numbers called the entries of the matrix
You can also think of a matrix as a list of row vectors or column vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
The length of a vector v in R2 and R3 is commonly denoted by the symbol ||v||
From the theorem of Pythagoras
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Unit Vectors
A vector of length 1 is called a unit vector
Normalizing v
Example 2Example 2
Find the unit vector u that has the same direction as v=(22-1)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors
In R2 these vectors are denoted by
In R3 these vectors are denoted by
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Sums of Three or More Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Parallel and Collinear Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Linear Combinations
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Application to Computer Color Models
Colors on computer monitors are commonly based on what is called the RGB color model
Each color vector c in RGB space or the RGB color cube is expressible as a linear combination of the form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Alternative Notations for Vectors
Comma-delimited form
Row-vector form
Column-vector form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Matrices
We define a matrix to be a rectangular array of numbers called the entries of the matrix
You can also think of a matrix as a list of row vectors or column vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
The length of a vector v in R2 and R3 is commonly denoted by the symbol ||v||
From the theorem of Pythagoras
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Unit Vectors
A vector of length 1 is called a unit vector
Normalizing v
Example 2Example 2
Find the unit vector u that has the same direction as v=(22-1)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors
In R2 these vectors are denoted by
In R3 these vectors are denoted by
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Parallel and Collinear Vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Linear Combinations
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Application to Computer Color Models
Colors on computer monitors are commonly based on what is called the RGB color model
Each color vector c in RGB space or the RGB color cube is expressible as a linear combination of the form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Alternative Notations for Vectors
Comma-delimited form
Row-vector form
Column-vector form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Matrices
We define a matrix to be a rectangular array of numbers called the entries of the matrix
You can also think of a matrix as a list of row vectors or column vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
The length of a vector v in R2 and R3 is commonly denoted by the symbol ||v||
From the theorem of Pythagoras
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Unit Vectors
A vector of length 1 is called a unit vector
Normalizing v
Example 2Example 2
Find the unit vector u that has the same direction as v=(22-1)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors
In R2 these vectors are denoted by
In R3 these vectors are denoted by
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Linear Combinations
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Application to Computer Color Models
Colors on computer monitors are commonly based on what is called the RGB color model
Each color vector c in RGB space or the RGB color cube is expressible as a linear combination of the form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Alternative Notations for Vectors
Comma-delimited form
Row-vector form
Column-vector form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Matrices
We define a matrix to be a rectangular array of numbers called the entries of the matrix
You can also think of a matrix as a list of row vectors or column vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
The length of a vector v in R2 and R3 is commonly denoted by the symbol ||v||
From the theorem of Pythagoras
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Unit Vectors
A vector of length 1 is called a unit vector
Normalizing v
Example 2Example 2
Find the unit vector u that has the same direction as v=(22-1)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors
In R2 these vectors are denoted by
In R3 these vectors are denoted by
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Application to Computer Color Models
Colors on computer monitors are commonly based on what is called the RGB color model
Each color vector c in RGB space or the RGB color cube is expressible as a linear combination of the form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Alternative Notations for Vectors
Comma-delimited form
Row-vector form
Column-vector form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Matrices
We define a matrix to be a rectangular array of numbers called the entries of the matrix
You can also think of a matrix as a list of row vectors or column vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
The length of a vector v in R2 and R3 is commonly denoted by the symbol ||v||
From the theorem of Pythagoras
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Unit Vectors
A vector of length 1 is called a unit vector
Normalizing v
Example 2Example 2
Find the unit vector u that has the same direction as v=(22-1)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors
In R2 these vectors are denoted by
In R3 these vectors are denoted by
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Alternative Notations for Vectors
Comma-delimited form
Row-vector form
Column-vector form
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Matrices
We define a matrix to be a rectangular array of numbers called the entries of the matrix
You can also think of a matrix as a list of row vectors or column vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
The length of a vector v in R2 and R3 is commonly denoted by the symbol ||v||
From the theorem of Pythagoras
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Unit Vectors
A vector of length 1 is called a unit vector
Normalizing v
Example 2Example 2
Find the unit vector u that has the same direction as v=(22-1)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors
In R2 these vectors are denoted by
In R3 these vectors are denoted by
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
11 Vectors and Matrices11 Vectors and Matrices
Matrices
We define a matrix to be a rectangular array of numbers called the entries of the matrix
You can also think of a matrix as a list of row vectors or column vectors
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
The length of a vector v in R2 and R3 is commonly denoted by the symbol ||v||
From the theorem of Pythagoras
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Unit Vectors
A vector of length 1 is called a unit vector
Normalizing v
Example 2Example 2
Find the unit vector u that has the same direction as v=(22-1)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors
In R2 these vectors are denoted by
In R3 these vectors are denoted by
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
The length of a vector v in R2 and R3 is commonly denoted by the symbol ||v||
From the theorem of Pythagoras
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Unit Vectors
A vector of length 1 is called a unit vector
Normalizing v
Example 2Example 2
Find the unit vector u that has the same direction as v=(22-1)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors
In R2 these vectors are denoted by
In R3 these vectors are denoted by
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Norm of A Vector
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Unit Vectors
A vector of length 1 is called a unit vector
Normalizing v
Example 2Example 2
Find the unit vector u that has the same direction as v=(22-1)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors
In R2 these vectors are denoted by
In R3 these vectors are denoted by
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Unit Vectors
A vector of length 1 is called a unit vector
Normalizing v
Example 2Example 2
Find the unit vector u that has the same direction as v=(22-1)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors
In R2 these vectors are denoted by
In R3 these vectors are denoted by
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors
In R2 these vectors are denoted by
In R3 these vectors are denoted by
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Standard Unit Vectors
Standard unit vectors in Rn
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12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
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12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
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12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Distance between Points in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Dot Products
Example 3Example 3
International Standard Book Number or ISBN
0-471-15307-9
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Algebraic Properties of The Dot Products
Example 4Example 4
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
The angle between u and v the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other Algebraically the radian measure of is in the interval 0leθleπ and in R2 the angle is generated by a counterclockwise rotation
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Angle Between Vectors in R2 and R3
Example 6Example 6
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthogonality
Example 7Example 7
If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn
wv=0 for every vector v in Rn w=0
Example 8Example 8
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Orthonormal Sets
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
12 Dot Product and Orthogonality12 Dot Product and Orthogonality
Euclidean Geometry in Rn
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Vector And Parametric Equations of Lines
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines Through Two Points
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 1Example 1
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Point-Normal Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 3Example 3
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 4Example 4
Example 5Example 5A plane is uniquely determined by three noncollinear points
Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Vector and Parametric Equations of Planes
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
Example 6Example 6Find a vector equation of the plane whose parametric equations are
Example 7Example 7
Find parametric equations of the plane x-y+2z=5
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Lines and Planes in Rn
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin
E-mail hogijunghanyangackrhttpwebyonseiackrhgjung
Comments on Terminology
13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes
A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin