Multivariable Calculus
Multivariable Calculus
11th Edition
ISBN: 9781337275378
Author: Ron Larson, Bruce H. Edwards
Publisher: Cengage Learning
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Textbook Question
Chapter 11, Problem 1RE

Writing Vectors in Different Forms In Exercises 1 and 2, let u = P Q and v = P R and (a) write u and v in component form, (b) write u and v as the linear combination of the standard unit vectors i and j, (c) find the magnitudes of u and v, and (d) find 3 u + v .

P = ( 1 , 2 ) , Q = ( 4 , 1 ) , R = ( 5 , 4 )

(a)

Expert Solution
Check Mark
To determine

To calculate: For given P=(1,2),Q=(4,1),R=(5,4), u=PQ,v=PR Find acomponent form of u and v.

Answer to Problem 1RE

Solution:

For given P=(1,2),Q=(4,1),R=(5,4), u=PQ,v=PR

Find a component form of u and vis 3,1 and 4,2 respectively.

Explanation of Solution

Given:

P=(1,2),Q=(4,1),R=(5,4), u=PQ,v=PR

Formula used:

If P(p1,p2) and Q(q1,q2) are the initial and terminal points of a directed line segment, then the component form of the vector u represented by PQ is

u1,u2=q1p1,q2p2

Calculation:

If

u2=q2p2=12=1

And

u =PQ and P=(1,2),Q=(4,1), then:

u1=q1p1=41=3

The component form of u is 3,1.

If v =PR and P=(1,2),R=(5,4), then:

v2=r2p2=42=2

And

v1=r1p1=51=4

The component form of v is 4,2.

(b)

Expert Solution
Check Mark
To determine

To calculate: For given P=(1,2),Q=(4,1),R=(5,4), u=PQ,v=PR, find u and vas the linear combination of the standard unit vectors i and j.

Answer to Problem 1RE

Solution: For given P=(1,2),Q=(4,1),R=(5,4), u=PQ,v=PR, u and vas the linear combination of the standard unit vectors i and j are the vectors u=3ij and v=4i+2j

Explanation of Solution

Given:

P=(1,2),Q=(4,1),R=(5,4), u=PQ,v=PR

Formula used:

If v1,v2 is the component form of v, then the vector v=v1i+v2j is called a linear combination of i and j.

Calculation:

The vectors u=3ij and v=4i+2j are the linear combinations of i and j.

As according to the calculation of part (a), a component form of u and vis 3,1 and 4,2.

(c)

Expert Solution
Check Mark
To determine

To calculate: Forgiven P=(1,2),Q=(4,1),R=(5,4), u=PQ,v=PR. Find magnitudes of u and v.

Answer to Problem 1RE

Solution:

Forgiven P=(1,2),Q=(4,1),R=(5,4), u=PQ,v=PR

Magnitudes of u and vu is u=10 and magnitude of v is v=25.

Explanation of Solution

Given:

P=(1,2),Q=(4,1),R=(5,4), u=PQ,v=PR

Formula used:

According to the Distance Formula, the length (or magnitude) of vectoru is:

u=(q1p1)2+(q2p2)2=u12+u22

Calculation:

As per part (a),

v=4,2. The length of vector v is:

v=v12+v22=42+22=16+4=20

u=3,1. The length of vector u is:

u=u12+u22=32+(1)2=9+1=10

and

Therefore, the magnitude of u is u=10 and magnitude of v is v=25.

(d)

Expert Solution
Check Mark
To determine

To calculate: For given P=(1,2),Q=(4,1),R=(5,4), u=PQ,v=PR. Find the value of -3u+v.

Answer to Problem 1RE

Solution:

For given P=(1,2),Q=(4,1),R=(5,4), u=PQ,v=PR. The value of -3u+v is 5,5.

Explanation of Solution

Given:

P=(1,2),Q=(4,1),R=(5,4), u=PQ,v=PR

Formula used:

The vector sum of u and v is the vector:

u+v=u1+v1,u2+v2

The scalar multiple of c and u is the vector:

cu=cu1,cu2

Calculation:

Value of -3u+v using scalar multiplication and vector sum formulae:

As according to the calculation of part (a), a component form of u and v is 3,1 and 4,2.

-3u+v=33,1+4,2=9,3+4,2=5,5

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Chapter 11 Solutions

Multivariable Calculus

Ch. 11.1 - Writing a Vector in Different Forms In Exercises...Ch. 11.1 - Writing a Vector in Different Forms to Exercises...Ch. 11.1 - Prob. 13ECh. 11.1 - Writing a Vector in Different Forms to Exercises...Ch. 11.1 - Writing a Vector in Different Forms In Exercises...Ch. 11.1 - Writing a Vector in Different Forms In Exercises...Ch. 11.1 - Finding a Terminal Point In Exercise 17 and 18,...Ch. 11.1 - Finding a Terminal Point In Exercise 17 and 18,...Ch. 11.1 - Prob. 19ECh. 11.1 - Prob. 20ECh. 11.1 - Prob. 21ECh. 11.1 - Finding a Magnitude of a VectorIn Exercises 1924,...Ch. 11.1 - Finding a Magnitude of a VectorIn Exercises 1924,...Ch. 11.1 - Finding a Magnitude of a VectorIn Exercises 1924,...Ch. 11.1 - Sketching Scalar MultipliesIn Exercises 25 and 26,...Ch. 11.1 - Sketching Scalar MultipliesIn Exercises 25 and 26,...Ch. 11.1 - Using Vector Operations In Exercise 27 and 28, And...Ch. 11.1 - Using Vector Operations In Exercise 27 and 28, And...Ch. 11.1 - Sketching a Vector In Exercises 29-34, use the...Ch. 11.1 - Prob. 30ECh. 11.1 - Sketching a Vector In Exercises 29-34, use the...Ch. 11.1 - Sketching a Vector In Exercises 29-34, use the...Ch. 11.1 - Sketching a Vector In Exercises 29-34, use the...Ch. 11.1 - Prob. 34ECh. 11.1 - Finding a Unit Vector In Exercises 35-38, find the...Ch. 11.1 - Finding a Unit Vector In Exercises 35-38, find the...Ch. 11.1 - Finding a Unit Vector In Exercises 35-38, find the...Ch. 11.1 - Finding a Unit Vector In Exercises 35-38, find the...Ch. 11.1 - Finding MagnitudesIn Exercises 3942, find the...Ch. 11.1 - Prob. 40ECh. 11.1 - Finding MagnitudesIn Exercises 3942, find the...Ch. 11.1 - Prob. 42ECh. 11.1 - Using the Triangle Inequality In Exercises 43 und...Ch. 11.1 - Prob. 44ECh. 11.1 - Prob. 45ECh. 11.1 - Finding a Vector In Exercises 45-48, find the...Ch. 11.1 - Prob. 47ECh. 11.1 - Prob. 48ECh. 11.1 - Prob. 49ECh. 11.1 - Prob. 50ECh. 11.1 - Prob. 51ECh. 11.1 - Prob. 52ECh. 11.1 - Finding a Vector In Exercises 53-56, find the...Ch. 11.1 - Finding a Vector In Exercises 53-56, find the...Ch. 11.1 - Finding a Vector In Exercises 53-56, find the...Ch. 11.1 - Finding a Vector In Exercises 53-56, find the...Ch. 11.1 - Prob. 57ECh. 11.1 - Prob. 58ECh. 11.1 - Prob. 59ECh. 11.1 - HOW DO YOU SEE IT? 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Using the Three-Dimensional Coordinate System In...Ch. 11.2 - Using the Three-Dimensional Coordinate System In...Ch. 11.2 - Using the Three-Dimensional Coordinate System In...Ch. 11.2 - Prob. 22ECh. 11.2 - Using the Three-Dimensional Coordinate System In...Ch. 11.2 - Prob. 24ECh. 11.2 - Prob. 25ECh. 11.2 - Prob. 26ECh. 11.2 - Prob. 27ECh. 11.2 - Prob. 28ECh. 11.2 - Classifying a TriangleIn Exercises 2932, find the...Ch. 11.2 - Classifying a TriangleIn Exercises 2932, find the...Ch. 11.2 - Classifying a TriangleIn Exercises 2932, find the...Ch. 11.2 - Prob. 32ECh. 11.2 - Prob. 33ECh. 11.2 - Prob. 34ECh. 11.2 - Prob. 35ECh. 11.2 - Prob. 36ECh. 11.2 - Finding the Equation of a Sphere In Exercises...Ch. 11.2 - Prob. 38ECh. 11.2 - Finding the Equation of a SphereIn Exercises 3742,...Ch. 11.2 - Prob. 40ECh. 11.2 - Prob. 41ECh. 11.2 - Prob. 42ECh. 11.2 - Prob. 43ECh. 11.2 - Prob. 44ECh. 11.2 - Prob. 45ECh. 11.2 - Prob. 46ECh. 11.2 - Prob. 47ECh. 11.2 - Prob. 48ECh. 11.2 - Prob. 49ECh. 11.2 - 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Prob. 23ECh. 11.4 - Prob. 24ECh. 11.4 - Prob. 25ECh. 11.4 - Prob. 26ECh. 11.4 - Torque The brakes on a bicycle are applied using a...Ch. 11.4 - Prob. 28ECh. 11.4 - Prob. 29ECh. 11.4 - Prob. 30ECh. 11.4 - Finding a Triple Scalar Product In Exercises...Ch. 11.4 - Prob. 32ECh. 11.4 - Prob. 33ECh. 11.4 - Prob. 34ECh. 11.4 - Volume In Exercises 35 and 36, use t triple scalar...Ch. 11.4 - Prob. 36ECh. 11.4 - Volume In Exercises 37 and 38, find the volume of...Ch. 11.4 - Prob. 38ECh. 11.4 - EXPLORING CONCEPTS Comparing Dot Products Identify...Ch. 11.4 - Prob. 40ECh. 11.4 - EXPLORING CONCEPTS Cross ProductTwo nonzero...Ch. 11.4 - Prob. 42ECh. 11.4 - Prob. 43ECh. 11.4 - Prob. 44ECh. 11.4 - Prob. 45ECh. 11.4 - Prob. 46ECh. 11.4 - Prob. 47ECh. 11.4 - Prob. 48ECh. 11.4 - Prob. 49ECh. 11.4 - Prob. 50ECh. 11.4 - Prob. 51ECh. 11.4 - Prob. 52ECh. 11.4 - Prob. 53ECh. 11.4 - Proof Prove that u(vw)=(uw)v(uv)w.Ch. 11.4 - Prob. 55ECh. 11.5 - CONCEPT CHECK Parametric and Symmetric...Ch. 11.5 - Prob. 2ECh. 11.5 - Prob. 3ECh. 11.5 - Prob. 4ECh. 11.5 - Checking Points on a Line In Exercises 5 and 6,...Ch. 11.5 - Prob. 6ECh. 11.5 - Prob. 7ECh. 11.5 - Prob. 8ECh. 11.5 - Prob. 9ECh. 11.5 - Prob. 10ECh. 11.5 - Prob. 11ECh. 11.5 - Finding Parametric and Symmetric EquationsIn...Ch. 11.5 - Finding Parametric and Symmetric EquationsIn...Ch. 11.5 - Prob. 14ECh. 11.5 - Prob. 15ECh. 11.5 - Finding Parametric and Symmetric Equations In...Ch. 11.5 - Prob. 17ECh. 11.5 - Prob. 18ECh. 11.5 - Prob. 19ECh. 11.5 - Prob. 20ECh. 11.5 - Prob. 21ECh. 11.5 - Prob. 22ECh. 11.5 - Prob. 23ECh. 11.5 - Prob. 24ECh. 11.5 - Prob. 25ECh. 11.5 - Prob. 26ECh. 11.5 - Prob. 27ECh. 11.5 - Using Parametric and Symmetric EquationsIn...Ch. 11.5 - Prob. 29ECh. 11.5 - Prob. 30ECh. 11.5 - Prob. 31ECh. 11.5 - Prob. 32ECh. 11.5 - Finding a Point of IntersectionIn Exercises 3336,...Ch. 11.5 - Prob. 34ECh. 11.5 - Finding a Point of IntersectionIn Exercises 3336,...Ch. 11.5 - Prob. 36ECh. 11.5 - Prob. 37ECh. 11.5 - Checking Points in a Plane In Exercises 37 and 38,...Ch. 11.5 - Finding an Equation of a PlaneIn Exercises 3944,...Ch. 11.5 - Prob. 40ECh. 11.5 - Prob. 41ECh. 11.5 - Prob. 42ECh. 11.5 - Finding an Equation of a PlaneIn Exercises 3944,...Ch. 11.5 - Prob. 44ECh. 11.5 - Finding an Equation of a PlaneIn Exercises 4556,...Ch. 11.5 - Finding an Equation of a PlaneIn Exercises 4556,...Ch. 11.5 - Finding an Equation of a PlaneIn Exercises 4556,...Ch. 11.5 - Finding an Equation of a PlaneIn Exercises 4556,...Ch. 11.5 - Finding an Equation of a PlaneIn Exercises 4556,...Ch. 11.5 - Prob. 50ECh. 11.5 - Finding an Equation of a PlaneIn Exercises 4556,...Ch. 11.5 - Prob. 52ECh. 11.5 - Finding an Equation of a PlaneIn Exercises 4556,...Ch. 11.5 - Prob. 54ECh. 11.5 - Prob. 55ECh. 11.5 - Prob. 56ECh. 11.5 - Prob. 57ECh. 11.5 - Prob. 58ECh. 11.5 - Finding an Equation of a PlaneIn Exercises 5760,...Ch. 11.5 - Prob. 60ECh. 11.5 - Parallel PlanesIn Exercises 6164, determine...Ch. 11.5 - Prob. 62ECh. 11.5 - Prob. 63ECh. 11.5 - Prob. 64ECh. 11.5 - Intersection of PlanesIn Exercises 6568, (a) find...Ch. 11.5 - 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In Exercises 113118, determine...Ch. 11.5 - Prob. 115ECh. 11.5 - Prob. 116ECh. 11.5 - Prob. 117ECh. 11.5 - Prob. 118ECh. 11.6 - CONCEPT CHECK Quadric Surfaces How are quadric...Ch. 11.6 - Prob. 2ECh. 11.6 - Prob. 3ECh. 11.6 - CONCEPT CHECK Think About It Does every...Ch. 11.6 - Prob. 5ECh. 11.6 - Matching In Exercises 5-10, match the equation...Ch. 11.6 - Prob. 7ECh. 11.6 - Matching In Exercises 5-10, match the equation...Ch. 11.6 - Prob. 9ECh. 11.6 - Matching In Exercises 5-10, match the equation...Ch. 11.6 - Sketching a Surface in SpaceIn Exercises 1114,...Ch. 11.6 - Prob. 12ECh. 11.6 - Prob. 13ECh. 11.6 - Prob. 14ECh. 11.6 - Prob. 15ECh. 11.6 - Prob. 16ECh. 11.6 - Prob. 17ECh. 11.6 - Prob. 18ECh. 11.6 - Prob. 19ECh. 11.6 - Prob. 20ECh. 11.6 - Prob. 21ECh. 11.6 - Prob. 22ECh. 11.6 - Prob. 23ECh. 11.6 - Prob. 24ECh. 11.6 - Prob. 25ECh. 11.6 - Prob. 26ECh. 11.6 - Prob. 27ECh. 11.6 - Prob. 28ECh. 11.6 - Prob. 29ECh. 11.6 - Prob. 30ECh. 11.6 - Prob. 31ECh. 11.6 - Finding an Equation for a Surface of RevolutionIn...Ch. 11.6 - Prob. 33ECh. 11.6 - Prob. 34ECh. 11.6 - Prob. 35ECh. 11.6 - Prob. 36ECh. 11.6 - Finding a Generating CurveIn Exercises 3740, find...Ch. 11.6 - Prob. 38ECh. 11.6 - Prob. 39ECh. 11.6 - Prob. 40ECh. 11.6 - Prob. 41ECh. 11.6 - Prob. 42ECh. 11.6 - Prob. 43ECh. 11.6 - Analyzing a TraceIn Exercises 43 and 44, analyze...Ch. 11.6 - Prob. 45ECh. 11.6 - Prob. 46ECh. 11.6 - Prob. 47ECh. 11.6 - Prob. 48ECh. 11.6 - Using a Hyperbolic ParaboloidDetermine the...Ch. 11.6 - Prob. 50ECh. 11.6 - Prob. 51ECh. 11.7 - CONCEPT CHECK Cylindrical CoordinatesDescribe the...Ch. 11.7 - Prob. 2ECh. 11.7 - Prob. 3ECh. 11.7 - Prob. 4ECh. 11.7 - Prob. 5ECh. 11.7 - Prob. 6ECh. 11.7 - Prob. 7ECh. 11.7 - Prob. 8ECh. 11.7 - Prob. 9ECh. 11.7 - Prob. 10ECh. 11.7 - Prob. 11ECh. 11.7 - Prob. 12ECh. 11.7 - Prob. 13ECh. 11.7 - Prob. 14ECh. 11.7 - Rectangular-to-Cylindrical ConversionIn Exercises...Ch. 11.7 - Prob. 16ECh. 11.7 - Prob. 17ECh. 11.7 - Prob. 18ECh. 11.7 - Rectangular-to-Cylindrical ConversionIn Exercises...Ch. 11.7 - Prob. 20ECh. 11.7 - Prob. 21ECh. 11.7 - Prob. 22ECh. 11.7 - Prob. 23ECh. 11.7 - Prob. 24ECh. 11.7 - Prob. 25ECh. 11.7 - Prob. 26ECh. 11.7 - Prob. 27ECh. 11.7 - Prob. 28ECh. 11.7 - Prob. 29ECh. 11.7 - Prob. 30ECh. 11.7 - Prob. 31ECh. 11.7 - Prob. 32ECh. 11.7 - Prob. 33ECh. 11.7 - Prob. 34ECh. 11.7 - Prob. 35ECh. 11.7 - Prob. 36ECh. 11.7 - Prob. 37ECh. 11.7 - Prob. 38ECh. 11.7 - Prob. 39ECh. 11.7 - Prob. 40ECh. 11.7 - Prob. 41ECh. 11.7 - Prob. 42ECh. 11.7 - Rectangular-to-Spherical ConversionIn Exercises...Ch. 11.7 - Prob. 44ECh. 11.7 - Prob. 45ECh. 11.7 - Prob. 46ECh. 11.7 - Prob. 47ECh. 11.7 - Prob. 48ECh. 11.7 - Prob. 49ECh. 11.7 - Prob. 50ECh. 11.7 - Spherical-to-Rectangular Conversion In Exercises...Ch. 11.7 - Prob. 52ECh. 11.7 - Prob. 53ECh. 11.7 - Prob. 54ECh. 11.7 - Spherical-to-Rectangular Conversion In Exercises...Ch. 11.7 - Prob. 56ECh. 11.7 - Spherical-to-Rectangular Conversion In Exercises...Ch. 11.7 - Prob. 58ECh. 11.7 - Prob. 59ECh. 11.7 - Prob. 60ECh. 11.7 - Prob. 61ECh. 11.7 - Prob. 62ECh. 11.7 - Prob. 63ECh. 11.7 - Prob. 64ECh. 11.7 - Prob. 65ECh. 11.7 - Prob. 66ECh. 11.7 - Prob. 67ECh. 11.7 - Prob. 68ECh. 11.7 - Prob. 69ECh. 11.7 - Prob. 70ECh. 11.7 - Prob. 71ECh. 11.7 - Prob. 72ECh. 11.7 - Prob. 73ECh. 11.7 - MatchingIn Exercises 7176, match the equation...Ch. 11.7 - MatchingIn Exercises 7176, match the equation...Ch. 11.7 - Prob. 76ECh. 11.7 - Prob. 77ECh. 11.7 - Prob. 78ECh. 11.7 - Prob. 79ECh. 11.7 - Prob. 80ECh. 11.7 - Prob. 81ECh. 11.7 - Prob. 82ECh. 11.7 - Converting a Rectangular EquationIn Exercises...Ch. 11.7 - Prob. 84ECh. 11.7 - Prob. 85ECh. 11.7 - Prob. 86ECh. 11.7 - Sketching a Solid In Exercises 8790, sketch the...Ch. 11.7 - Prob. 88ECh. 11.7 - Sketching a SolidIn Exercises 8790, sketch the...Ch. 11.7 - Prob. 90ECh. 11.7 - Prob. 91ECh. 11.7 - Prob. 92ECh. 11.7 - Prob. 93ECh. 11.7 - Prob. 94ECh. 11.7 - Prob. 95ECh. 11.7 - Prob. 96ECh. 11.7 - Prob. 97ECh. 11.7 - Prob. 98ECh. 11.7 - Prob. 99ECh. 11.7 - Prob. 100ECh. 11.7 - Prob. 101ECh. 11.7 - Prob. 102ECh. 11.7 - Intersection of SurfaceIdentify the curve of...Ch. 11.7 - Prob. 104ECh. 11 - Writing Vectors in Different Forms In Exercises 1...Ch. 11 - Prob. 2RECh. 11 - Prob. 3RECh. 11 - Prob. 4RECh. 11 - Prob. 5RECh. 11 - Prob. 6RECh. 11 - Prob. 7RECh. 11 - Prob. 8RECh. 11 - Prob. 9RECh. 11 - Prob. 10RECh. 11 - Prob. 11RECh. 11 - Prob. 12RECh. 11 - Prob. 13RECh. 11 - Prob. 14RECh. 11 - Prob. 15RECh. 11 - Prob. 16RECh. 11 - Prob. 17RECh. 11 - Prob. 18RECh. 11 - Prob. 19RECh. 11 - Prob. 20RECh. 11 - Prob. 21RECh. 11 - Prob. 22RECh. 11 - Prob. 23RECh. 11 - Finding the Angle Between Two Vectors In Exercises...Ch. 11 - Prob. 25RECh. 11 - Prob. 26RECh. 11 - Prob. 27RECh. 11 - Finding the Projection of u onto v In Exercises 27...Ch. 11 - Prob. 29RECh. 11 - Prob. 30RECh. 11 - Prob. 31RECh. 11 - Prob. 32RECh. 11 - Finding a Unit VectorFind a unit vector that is...Ch. 11 - AreaFind the area of the parallelogram that has...Ch. 11 - Prob. 35RECh. 11 - VolumeUse the triple scalar product to find the...Ch. 11 - Finding Parametric and Symmetric Equations In...Ch. 11 - Prob. 38RECh. 11 - Prob. 39RECh. 11 - Prob. 40RECh. 11 - Prob. 41RECh. 11 - Prob. 42RECh. 11 - Finding an Equation of a Plane In Exercises 41-44,...Ch. 11 - Prob. 44RECh. 11 - Prob. 45RECh. 11 - Prob. 46RECh. 11 - Distance Find the distance between the planes...Ch. 11 - Distance Find the distance between the point...Ch. 11 - Prob. 49RECh. 11 - Prob. 50RECh. 11 - Prob. 51RECh. 11 - Prob. 52RECh. 11 - Prob. 53RECh. 11 - Prob. 54RECh. 11 - Prob. 55RECh. 11 - Prob. 56RECh. 11 - Prob. 57RECh. 11 - Prob. 58RECh. 11 - Prob. 59RECh. 11 - Prob. 60RECh. 11 - Prob. 61RECh. 11 - Prob. 62RECh. 11 - Prob. 63RECh. 11 - Cylindrical-to-Rectangular ConversionIn Exercises...Ch. 11 - Prob. 65RECh. 11 - Spherical-to-Rectangular ConversionIn Exercises 65...Ch. 11 - Converting a Rectangular EquationIn Exercises 67...Ch. 11 - Prob. 68RECh. 11 - Cylindrical-to-Rectangular Conversion In Exercises...Ch. 11 - Cylindrical-to- Rectangular ConversionIn Exercises...Ch. 11 - Prob. 71RECh. 11 - Spherical-to-Rectangular Conversion In Exercises...Ch. 11 - ProofUsing vectors, prove the Law of Sines: If a,...Ch. 11 - Prob. 2PSCh. 11 - Prob. 3PSCh. 11 - Proof Using vectors, prove that the diagonals of a...Ch. 11 - Distance (a) Find the shortest distance between...Ch. 11 - Prob. 6PSCh. 11 - Volume (a) Find the volume of the solid bounded...Ch. 11 - Prob. 8PSCh. 11 - Prob. 9PSCh. 11 - Prob. 10PSCh. 11 - Prob. 11PSCh. 11 - Prob. 12PSCh. 11 - Prob. 13PSCh. 11 - Prob. 14PSCh. 11 - Prob. 15PSCh. 11 - Prob. 16PSCh. 11 - Distance Between a Point and a PlaneConsider the...Ch. 11 - Prob. 18PSCh. 11 - Prob. 19PS
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