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Transcript of Chapter 9-Vectors Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All...
Chapter 9-Vectors
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Chapter 9-Vectors9.1 Vectors in the Plane
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
INSERT FIGURE 9-1-1
Chapter 9-Vectors9.1 Vectors in the Plane
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Chapter 9-Vectors9.1 Vectors in the Plane
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Vectors
Chapter 9-Vectors9.1 Vectors in the Plane
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
DEFINTION: The sum v+w of two vectors v=<v1,v2> and w=<w1,w2> is formed by adding the vectors componentwise:
Vector Algebra
Chapter 9-Vectors9.1 Vectors in the Plane
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
EXAMPLE: Add the vectors v = <−3, 9> and w = <1, 8>.
Vector Algebra
DEFINITION: The zero vector 0 is the vector both of whose components are 0.
DEFINITION: If v = <v1, v2> is a vector and is a real number then we define the scalar multiplication of v by tobe
Chapter 9-Vectors9.1 Vectors in the Plane
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Length (or Magnitude) of a Vector
THEOREM: If v is a vector and is a scalar, then
Chapter 9-Vectors9.1 Vectors in the Plane
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Unit Vectors and Directions
Chapter 9-Vectors9.1 Vectors in the Plane
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Unit Vectors and Directions
DEFINITION: Let v and w be nonzero vectors. We say that v and w have the same direction if dir(v) =dir(w). We say that v and w are opposite in direction if dir(v) = −dir(w). We say that v and w are parallel if either (i) v and w have the same direction or (ii) v and w are opposite in direction. Although the zero vector 0 does not have a direction, it is conventional to say that 0 is parallel to every vector.
Chapter 9-Vectors9.1 Vectors in the Plane
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Unit Vectors and Directions
THEOREM: Vectors v and w are parallel if and only if at least one of the following two equations holds: (i) v = 0 or (ii) w = v for some scalar . Moreover, if v and w are both nonzero and w = v, then v and w have the same direction if 0 < and opposite directions if < 0.
Chapter 9-Vectors9.1 Vectors in the Plane
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Unit Vectors and Directions
EXAMPLE: For what value of a are the vectors <a,−1> and <3, 4> parallel?
Chapter 9-Vectors9.1 Vectors in the Plane
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
An Application to Physics
EXAMPLE: Two workers are each pulling on a rope attached to a dead tree stump. One pulls in the northerly direction with a force of 100 pounds and the other in the easterly direction with a force of 75 pounds. Compute the resultant force that is applied to the tree stump.
Chapter 9-Vectors9.1 Vectors in the Plane
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Special Unit Vectors i and j
i=<1,0> and j=<0,1>
Suppose that v=<3,-5> and w=<2,4>. Express v, w, and v+w in terms of i and j.
Chapter 9-Vectors9.1 Vectors in the Plane
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Triangle Inequality
EXAMPLE: Verify the Triangle Inequality for the vectors v = <−3, 4> and w = <8, 6>.
Chapter 9-Vectors9.1 Vectors in the Plane
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Quick Quiz
Chapter 9-Vectors9.2 Vectors in Three-Dimensional Space
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
EXAMPLE: Sketch the points (3,2,5), (2,3,-3), and (-1,-2,1).
Chapter 9-Vectors9.2 Vectors in Three-Dimensional Space
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Distance
THEOREM: The distance d(P,Q) between points P=(p1, p2, p3) and Q=(q1, q2, q3) is given by
Chapter 9-Vectors9.2 Vectors in Three-Dimensional Space
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Distance
EXAMPLE: Determine what set of points is described by the equation
Chapter 9-Vectors9.2 Vectors in Three-Dimensional Space
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Distance
DEFINITION: Let P0 = (x0, y0, z0) be a point in space and let r be a positive number. The set
is the set of all points inside the sphere
This set is called the open ball with center P0 and radius r.
Chapter 9-Vectors9.2 Vectors in Three-Dimensional Space
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Vectors in Space
Chapter 9-Vectors9.2 Vectors in Three-Dimensional Space
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Vector Operations
EXAMPLE: Suppose v=<3,0,1> and w=<0,4,2>. Calculate v+w, and sketch the three vectors.
EXAMPLE: Suppose v=<2,-1,1>. Calculate 3v and -4v.
Chapter 9-Vectors9.2 Vectors in Three-Dimensional Space
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Length of a Vector
If v=<v1,v2,v3> is a vector, then the length or magnitude of v is defined to be
Chapter 9-Vectors9.2 Vectors in Three-Dimensional Space
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Unit Vectors and Directions
EXAMPLE: Is there a value of r for which u=<-1/3,2/3,r> is a unit vector?
EXAMPLE: Suppose that v=<4,3,1> and w=<2,b,c>. Are there values of b and c for which v and w are parallel?
Chapter 9-Vectors9.2 Vectors in Three-Dimensional Space
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Quick Quiz
Chapter 9-Vectors9.3 The Dot Product and Applications
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Algebraic Definition of the Dot Product
DEFINITION: The dot product vw of two vectors v and w is the sum of the products of corresponding entries of v and w.
EXAMPLE: Let u = <2, 3,−1>, v = <4, 6,−2>, and w = <−2,−1,−7>. Calculate the dot products u · v, u · w, and v · w.
Chapter 9-Vectors9.3 The Dot Product and Applications
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Algebraic Definition of the Dot Product
THEOREM: Suppose that u, v, and w are vectors and that is a scalar. The dot product satisfies the following elementary properties:
Chapter 9-Vectors9.3 The Dot Product and Applications
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Algebraic Definition of the Dot Product
EXAMPLE: Let u=<3,2> and v=<4,-5>. Calculate (u v)u+(v u)v
Chapter 9-Vectors9.3 The Dot Product and Applications
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
A Geometric Formula for the Dot Product
THEOREM: Let v and w be nonzero vectors. Then the angle between v and w satisfies the equation
Chapter 9-Vectors9.3 The Dot Product and Applications
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
A Geometric Formula for the Dot Product
EXAMPLE: Calculate the angle between the two vectors v=<2,2,4>and w=<2,-1,1>.
Chapter 9-Vectors9.3 The Dot Product and Applications
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
A Geometric Formula for the Dot Product
Cauchy-Schwarz Inequality:
EXAMPLE: Verify that the two vectors v = <2, 2, 4> and w = <12, 13, 24> satisfy the Cauchy-Schwarz Inequality.
Chapter 9-Vectors9.3 The Dot Product and Applications
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
A Geometric Formula for the Dot Product
DEFINTION: Let be the angle between nonzero vectors v and w. If = /2 then we say that the vectors v and w are orthogonal or mutually perpendicular.
Chapter 9-Vectors9.3 The Dot Product and Applications
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
A Geometric Formula for the Dot Product
THEOREM: Let v and w be any vectors. Then:a) v and w are orthogonal if and only if v · w = 0.
b) v and w are parallel if and only if
c) If v and w are nonzero, and if is the angle between them, then v and w are parallel if and only if = 0or = . In this case, v and w have the same direction if = 0 and opposite directions if = .
Chapter 9-Vectors9.3 The Dot Product and Applications
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
A Geometric Formula for the Dot Product
EXAMPLE: Consider the vectors u = <2, 3,−1>, v = <4, 6,−2>, and w = <−2,−1,−7>. Are any of these vectors orthogonal? Parallel?
Chapter 9-Vectors9.3 The Dot Product and Applications
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Projection
THEOREM: If v and w are nonzero vectors then the projection Pw(v) of v onto w is given by
The length of Pw(v) is given by
Chapter 9-Vectors9.3 The Dot Product and Applications
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Projection
EXAMPLE: Let v = <2, 1,−1> and w = <1,−2, 2>. Calculate the projection of v onto w, the projection of w onto v, and calculate the lengths of these projections.
Also calculate the component of v in the direction of w and the component of w in the direction of v.
Chapter 9-Vectors9.3 The Dot Product and Applications
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Projection and the Standard Basis Vectors
EXAMPLE: Let v = <2,-6,12>. Calculate Pi(v), Pj(v), and Pk(v) and express v as a linear combination of these projections.
Chapter 9-Vectors9.3 The Dot Product and Applications
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Direction Cosines and Direction Angles
EXAMPLE: Calculate the direction cosines and direction angels for the vector
Chapter 9-Vectors9.3 The Dot Product and Applications
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Applications
EXAMPLE: A tow truck pulls a disabled vehicle a total of 20,000 feet. In order to keep the vehicle in motion, the truck must apply a constant force of 3, 000 pounds. The hitch is set up so that the force is exerted at an angle of 30 with the horizontal. How much work is performed?
Chapter 9-Vectors9.3 The Dot Product and Applications
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Quick Quiz
1. Calculate <1, 2,−1> · <3, 4, 2>.
2. Use the arccosine to express the angle between <6, 3, 2> and <4,−7, 4>.
3. For what value of a are <1, a,−1> and <3, 4, a> perpendicular?
4. Calculate the projection of <3, 1,−1> onto <8, 4, 1>.
Chapter 9-Vectors9.4 The Cross Product and Triple Product
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Cross Product of Two Spatial Vectors
DEFINTION: If v = <v1, v2, v3> and w = <w1,w2,w3>, then we define their cross product v × w to be
Chapter 9-Vectors9.4 The Cross Product and Triple Product
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Cross Product of Two Spatial Vectors
THEOREM: If v and w are vectors, then v × w is orthogonal to both v and w.
EXAMPLE: Let v = <2,−1, 3> and w = <5, 4,−6>. Calculate v × w. Verify that v × w is orthogonal to both v and w.
Chapter 9-Vectors9.4 The Cross Product and Triple Product
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Relationship between Cross Products and Determinants
If v = <v1, v2, v3> and w = <w1,w2,w3>, then
EXAMPLE: Use a determinant to calculate the cross product of v = <2,−1, 6> and w = <−3, 4, 1>.
Chapter 9-Vectors9.4 The Cross Product and Triple Product
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Algebraic Properties of the Cross Product
If u, v, and w are vectors and and are scalars, then
Chapter 9-Vectors9.4 The Cross Product and Triple Product
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Algebraic Properties of the Cross Product
THEOREM: If v is any vector, then v×v = 0. More generally, if u and v are parallel vectors then u×v = 0.
EXAMPLE: Give an example to show that the cross product does not satisfy a cancellation property.
Give an example to show that the cross product does not satisfy the associative property.
Chapter 9-Vectors9.4 The Cross Product and Triple Product
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
A Geometric Understanding of the Cross Product
EXAMPLE: Find the standard unit normal for the pairs (i, j) and (j, k) and (k, i). Find also the standard unit normal for the pairs (j, i) and (k, j) and (i, k).
Chapter 9-Vectors9.4 The Cross Product and Triple Product
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
A Geometric Understanding of the Cross Product
THEOREM: Let v and w be vectors. Thena) b) If v and w are nonzero, thenwhere [0,] denotes the angle between v and w;c) v and w are parallel if and only if v × w = 0;d) If v and w are not parallel, then v × w points in the direction of the standard unit normal for the pair (v,w).
Chapter 9-Vectors9.4 The Cross Product and Triple Product
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
A Geometric Understanding of the Cross Product
EXAMPLE: Let v = <1,−3, 2> and w = <1,−1, 4>. What is the standard unit normal vector for (v,w)?
Chapter 9-Vectors9.4 The Cross Product and Triple Product
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Cross Products and the Calculation of Area
THEOREM: Suppose that v and w are nonparallel vectors. The area of the triangle determined by v and w is The area of the parallelogram determined by v and w is
EXAMPLE: Find the area of the parallelogram determined by the vectors v=<-2,1,3> and w=<1,0,4>.
Chapter 9-Vectors9.4 The Cross Product and Triple Product
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Triple Scalar Product
DEFINITION: If u, v, and w are given vectors, then we define their triple scalar product to be the number (u×v) · w.
EXAMPLE: Calculate the triple scalar product of u = <2,−1, 4>, v = <7, 2, 3>, and w = <−1, 1, 2> in two different ways.
Chapter 9-Vectors9.4 The Cross Product and Triple Product
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Triple Scalar Product
THEOREM: The triple scalar product ofu = <u1, u2, u3>, v = <v1, v2, v3>, and w = <w1,w2,w3> is given by the formula
Chapter 9-Vectors9.4 The Cross Product and Triple Product
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Triple Scalar Product
EXAMPLE: Use the determinant to calculate the volume of the parallelepiped determined by the vectors <−3, 2, 5>, <1, 0, 3>, and <3,−1,−2>.
Chapter 9-Vectors9.4 The Cross Product and Triple Product
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Triple Scalar Product
THEOREM: Three vectors u ,v, and w are coplanar if and only if u · (v × w) = 0.
Chapter 9-Vectors9.4 The Cross Product and Triple Product
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Triple Vector Products
DEFINITION: If u, v, and w are given spatial vectors, then each of the vectors u × (v × w) and (u × v) × w is said to be a triple vector product of u, v, and w.
EXAMPLE: Let v and w be perpendicular spatial vectors. Show that
and
Chapter 9-Vectors9.4 The Cross Product and Triple Product
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Quick Quiz
1. Calculate <2, 1, 2> × <1,−2,−1>.
2. Find the area of the parallelogram determined by <2, 1,−2> and <1, 1, 0>.
3. Find the standard unit normal vector for the ordered pair (<2, 1,−2>, <1, 1, 0>) .
4. True or false: a) v × w = w × v b) u × (v × w) = (u × v) × w c) u · v × w = u × v · w?
Chapter 9-Vectors9.5 Lines and Planes in Space
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Cartesian Equations of Planes in Space
THEOREM: Let V be a plane in space. Suppose that n=<A,B,C> is a normal vector for V and that P0=(x0,y0,z0) is a point on V. Then
is a Cartesian equation for V.
Chapter 9-Vectors9.5 Lines and Planes in Space
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Cartesian Equations of Planes in Space
THEOREM: Suppose at least one of the coefficients A, B, C is nonzero. Then the solution set of the equation
A(x−x0)+B(y−y0)+C(z−z0) = 0
is the plane that has <A,B,C> as a normal vector and passes through the point (x0, y0, z0). The solution set of the equation Ax + By + Cz = D is a plane that has <A,B,C> as a normal vector.
Chapter 9-Vectors9.5 Lines and Planes in Space
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Cartesian Equations of Planes in Space
EXAMPLE: Find an equation for the plane V passing through the points P = (2,−1, 4), Q = (3, 1, 2), and R = (6, 0, 5).
EXAMPLE: Find the angle between the plane with Cartesian equation x − y − z = 7 and the plane with Cartesian equation −x + y − 3z = 6.
Chapter 9-Vectors9.5 Lines and Planes in Space
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Parametric Equations of Planes in SpaceTHEOREM: If P0 = (x0, y0, z0) is a point on a plane V and if u = <u1, u2, u3> and v = <v1, v2, v3> are any two nonparallel vectors that are perpendicular to a normal vector for V, then V consists precisely of those points (x, y, z) with coordinates that satisfy the vector equation
When written coordinatewise, equation above yields parametric equations for V:
Chapter 9-Vectors9.5 Lines and Planes in Space
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Parametric Equations of Planes in Space
EXAMPLE: Find parametric equations for the plane V whose Cartesian equation is 3x − y + 2z = 7.
Chapter 9-Vectors9.5 Lines and Planes in Space
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Parametric Equations of Lines in SpaceTHEOREM: The line in space that passes through the point P0 = (x0, y0, z0) and is parallel to the vector m = <a, b, c> has equation
Here P = (x, y, z) is a variable point on the line. In coordinates the equation may be written as three parametric equations:
Chapter 9-Vectors9.5 Lines and Planes in Space
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Parametric Equations of Lines in Space
Chapter 9-Vectors9.5 Lines and Planes in Space
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Cartesian Equations of Line in Space
Chapter 9-Vectors9.5 Lines and Planes in Space
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Cartesian Equations of Line in Space
EXAMPLE: Find parametric equations of the line of intersection of the two planes
x − 2y + z = 4 and 2x + y − z = 3.
Chapter 9-Vectors9.5 Lines and Planes in Space
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Cartesian Equations of Line in Space
EXAMPLE: Find parametric equations of the line of intersection of the two planes
x − 2y + z = 4 and 2x + y − z = 3.
Chapter 9-Vectors9.5 Lines and Planes in Space
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Calculating Distance
THEOREM: Suppose that P = (x0, y0, z0) is a point and that V is a plane. Let n = <A,B,C> be a normal vector for V and let Q = (x1, y1, z1) be any point on V. The distance between P and V is equal to
Chapter 9-Vectors9.5 Lines and Planes in Space
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Calculating Distance
EXAMPLE: Find the distance between the point P = (3,−8, 3) and the plane V whose Cartesian equation is 2x + y − 2z = 10.
Chapter 9-Vectors9.5 Lines and Planes in Space
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Quick Quiz