Phy10 Vectors
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Transcript of Phy10 Vectors
Vectors
Engr. E. D. Dimaunahan
1. The system of units used by scientists and engineers around the world is commonly called?
2. Write the equivalent of the following Greek Prefixes?Micro, mili, kilo, nano
3. Differentiate a scalar and vector quantities.
4. Other name for vector sum.
5. Give examples of scalar and vector quantities.
Vectors and Their Components
Physics deals with quantities that have both size and direction
A vector is a mathematical object with size and direction
A vector quantity is a quantity that can be represented by a vector
Examples: position, velocity, acceleration Vectors have their own rules for manipulation
A scalar is a quantity that does not have a direction Examples: time, temperature, energy, mass Scalars are manipulated with ordinary algebra
VECTORS
resultant vector – the vector sum of two or more concurrent vectors.
equilibrant vector – a single vector that balances two or more concurrent vectors.
equivalent vectors – two vectors that have the same magnitude and direction.
parallel vectors – two vectors that have the same direction.
anti-parallel vectors – two vectors that have opposite directions.
negative of a vector – a vector that has the same magnitude as but opposite the direction of another vector.
Vectors and Their Components
The simplest example is a displacement vector
If a particle changes position from A to B, we represent this by a vector arrow pointing from A to B
In (a) we see that all three arrows have the same magnitude and direction: they are identical displacement vectors.
In (b) we see that all three paths correspond to the same displacement vector. The vector tells us nothing about the actual path that was taken between A and B.
Vectors and Their Components
The vector sum, or resultanto Is the result of performing vector additiono Represents the net displacement of two or more
displacement vectors
o Can be added graphically as shown:
72.4 m, 32 degrees east of north57.3 m, 36 degrees south of west17.8 m straight south
Vectors and Their Components
Vector addition is commutativeo We can add vectors in any order
Eq. (3-2)
Figure (3-3)
Vectors and Their Components
Vector addition is associativeo We can group vector addition however we like
Figure (3-4)
Vectors and Their Components
A negative sign reverses vector direction
We use this to define vector subtraction
Eq. (3-4)
Vectors and Their Components
These rules hold for all vectors, whether they represent displacement, velocity, etc.
Only vectors of the same kind can be addedo (distance) + (distance) makes senseo (distance) + (velocity) does not
Answer:
(a) 3 m + 4 m = 7 m (b) 4 m - 3 m = 1 m
Vectors and Their Components
Rather than using a graphical method, vectors can be added by componentso A component is the projection of a vector on an axis
The process of finding components is called resolving the vector
The components of a vector can be positive or negative.
They are unchanged if the vector is shifted in any direction (but not rotated).
Vectors and Their Components
Components in two dimensions can be found by:
Where θ is the angle the vector makes with the positive x axis, and a is the vector length
The length and angle can also be found if the components are known
Therefore, components fully define a vector
Vectors and Their Components
In the three dimensional case we need more components to specify a vector
o (a,θ,φ) or (ax,a
y,a
z)
Vectors and Their Components
Angles may be measured in degrees or radians
Recall that a full circle is 360˚, or 2π rad
Know the three basic trigonometric functions
Figure (3-11)
© 2014 John Wiley & Sons, Inc. All rights reserved.
Unit Vectors, Adding Vectors by Components
A unit vectoro Has magnitude 1o Has a particular directiono Lacks both dimension and unito Is labeled with a hat: ^
We use a right-handed coordinate systemo Remains right-handed when rotated
Unit Vectors, Adding Vectors by Components
The quantities axi and a
yj are vector components
The quantities ax and a
y alone are scalar components
o Or just “components” as before
Vectors can be added using components
Eq. (3-9) →
Unit Vectors, Adding Vectors by Components
To subtract two vectors, we subtract components
Unit Vectors, Adding Vectors by Components
Answer: (a) positive, positive (b) positive, negative
(c) positive, positive
Unit Vectors, Adding Vectors by Components
Vectors are independent of the coordinate system used to measure them
We can rotate the coordinate system, without rotating the vector, and the vector remains the same
All such coordinate systems are equally valid
Multiplying Vectors
Multiplying a vector z by a scalar co Results in a new vectoro Its magnitude is the magnitude of vector z times |c| o Its direction is the same as vector z, or opposite if c is
negativeo To achieve this, we can simply multiply each of the
components of vector z by c
To divide a vector by a scalar we multiply by 1/cExample Multiply vector z by 5
o z = -3 i + 5 jo 5 z = -15 i + 25 j
Multiplying Vectors
Multiplying two vectors: the scalar producto Also called the dot producto Results in a scalar, where a and b are magnitudes and φ is
the angle between the directions of the two vectors:
The commutative law applies, and we can do the dot product in component form
Multiplying Vectors
A dot product is: the product of the magnitude of one vector times the scalar component of the other vector in the direction of the first vector
Eq. (3-21)
Either projection of one vector onto the other can be used
To multiply a vector by the projection, multiply the magnitudes
© 2014 John Wiley & Sons, Inc. All rights reserved.
Multiplying Vectors
Answer: (a) 90 degrees (b) 0 degrees (c) 180 degrees
Multiplying Vectors
Multiplying two vectors: the vector producto The cross product of two vectors with magnitudes a & b,
separated by angle φ, produces a vector with magnitude:
o And a direction perpendicular to both original vectors
Direction is determined by the right-hand rule
Place vectors tail-to-tail, sweep fingers from the first to the second, and thumb points in the direction of the resultant vector
Eq. (3-24)
Multiplying Vectors
The upper shows vector a cross vector b, the lower shows vector b cross vector a
Multiplying Vectors
The cross product is not commutative
To evaluate, we distribute over components:
Therefore, by expanding:
Multiplying Vectors
Answer: (a) 0 degrees (b) 90 degrees
UNIT VECTOR
A unit vector is a vector with a magnitude of exactly 1 and points in a particular direction.
+x
+y
+z
i
j
k
A vector in its “regular” form or “magnitude-angle” form, such as
xabovemA o 30,00.10
can be expressed in “unit vector” form or “rectangular” form.
jmimA ˆ)00.5(ˆ)66.8(
+x
+y
+z
iA xˆ
kA zˆ
jA yˆ
A
VECTOR ADDITION
+x
+y
+z
B
iB xˆ
kB Zˆ
jB yˆ
kmjmimA ˆ)6(ˆ)8(ˆ)10(
kmjmimB ˆ)4(ˆ)12(ˆ)8(
kmjmimA ˆ)6(ˆ)8(ˆ)10(
kmjmimB ˆ)4(ˆ)12(ˆ)8(
kmjmimC ˆ)10(ˆ)20(ˆ)2(
BAC
Find the resultant of the following vectors.
kmjmimA ˆ)6(ˆ)8(ˆ)10(
kmjmimB ˆ)4(ˆ)12(ˆ)8(
VECTOR PRODUCTS: Scalar/Dot Product
kAjAiAA zyxˆˆˆ
kBjBiBB zyxˆˆˆ
Given the following vectors:
The scalar (or dot) product of these vectors is defined as:
cosABBA
Where: A is the magnitude of A
B is the magnitude of B
is the angle between and .A
B
VECTOR PRODUCTS: Scalar/Dot Product
cosABBA
10cos)1)(1(ˆˆ ii
090cos)1)(1(ˆˆ ji
090cos)1)(1(ˆˆ ki
090cos)1)(1(ˆˆ ij
10cos)1)(1(ˆˆ jj
090cos)1)(1(ˆˆ kj
090cos)1)(1(ˆˆ ik
090cos)1)(1(ˆˆ jk
10cos)1)(1(ˆˆ kk
VECTOR PRODUCTS: Scalar/Dot Product
kBjBiBkAjAiABA zyxzyxˆˆˆˆˆˆ
kBjBiBB zyxˆˆˆ
kAjAiAA zyxˆˆˆ
zzyyxx BABABABA
cosABBA
Find the dot product of the following vectors:
kmjmimA ˆ)6(ˆ)8(ˆ)10(
kmjmimB ˆ)4(ˆ)12(ˆ)8(
zzyyxx BABABABA
)]4)(6[()]12)(8[()]8)(10[( mmmmmmBA
222 249680 mmmBA
240mBA
Find the angle between the following vectors:
kmjmimA ˆ)6(ˆ)8(ˆ)10(
kmjmimB ˆ)4(ˆ)12(ˆ)8(
cosABBA
cos22420040
224200
40cos 1
o11.79
VECTOR PRODUCTS: vector/Cross Product
kAjAiAA zyxˆˆˆ
kBjBiBB zyxˆˆˆ
Given the following vectors:
The vector (or cross) product of these vectors is defined as:
sinABBA
Where: A is the magnitude of A
B is the magnitude of B
is the angle between and .A
B
VECTOR PRODUCTS: Vector/Cross Product
sinABBA
00sin)1)(1(ˆˆ ii
kji ˆ90sin)1)(1(ˆˆ
jki ˆ90sin)1)(1(ˆˆ
kij ˆ90sin)1)(1(ˆˆ
00sin)1)(1(ˆˆ jj
ikj ˆ90sin)1)(1(ˆˆ
jik ˆ90sin)1)(1(ˆˆ
ijk ˆ90sin)1)(1(ˆˆ
00sin)1)(1(ˆˆ kk
kBjBiBkAjAiABA zyxzyxˆˆˆˆˆˆ
kBjBiBB zyxˆˆˆ
kAjAiAA zyxˆˆˆ
VECTOR PRODUCTS: Vector/Cross Product
sinABBA
kBABAjBABAiBABABA xyyxzxxzyzzyˆˆˆ
Find the cross product of the following vectors:
kmjmimA ˆ)6(ˆ)8(ˆ)10(
kmjmimB ˆ)4(ˆ)12(ˆ)8(
kBABAjBABAiBABABA xyyxzxxzyzzyˆˆˆ
kjiBA ˆ)64(120ˆ4048ˆ7232
kmjmimBA ˆ184ˆ88ˆ40 222
PROBLEM:
PROBLEM:
In the figure at right, a cube is placed so that one corner (point O) is at the origin and three edges are along the +x +y, and +z axes of a coordinate system. Use unit vectors to find the (a) angle between the edge along the z-axis (line OQ) and the diagonal of the cube (line OP). (b) angle between the diagonal of a face (line OR) and line OP.