Math

CalculusCalculus: Early Transcendental Functions (MindTap Course List)Finding an Equation of a Surface of Revolution In Exercises 31–36, find an equation for the surface of revolution formed by revolving the curve in the indicated coordinate plane about the given axis. Equationof Curve CoordinatePlane Axis ofRevolution z = 3 y y z − plane y − axisBuyFind*launch*

6th Edition

Ron Larson + 1 other

Publisher: Cengage Learning

ISBN: 9781285774770

Chapter 11.6, Problem 32E

Textbook Problem

Finding an Equation of a Surface of Revolution In Exercises 31–36, find an equation for the surface of revolution formed by revolving the curve in the indicated coordinate plane about the given axis.

Equationof Curve CoordinatePlane Axis ofRevolution

Calculus: Early Transcendental Functions (MindTap Course List)

Ch. 11.1 - Sketching a Vector In Exercises 14, (a) find the...Ch. 11.1 - Sketching a Vector In Exercises 14, (a) find the...Ch. 11.1 - Sketching a Vector In Exercises 3 and 4, (a) find...Ch. 11.1 - Sketching a Vector In Exercises 3 and 4, (a) find...Ch. 11.1 - Equivalent Vectors In Exercises 5-8, find the...Ch. 11.1 - Equivalent Vectors In Exercises 5-8, find the...Ch. 11.1 - Equivalent Vectors In Exercises 5-8, find the...Ch. 11.1 - Equivalent Vectors In Exercises 5-8, find the...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 - Writing a Vector in Different Forms In 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 - Writing a Vector in Different Forms In 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 Exercises 17 and 18,...Ch. 11.1 - Finding a Terminal Point In Exercises 17 and 18,...Ch. 11.1 - Using Vector Operations In Exercises 19 and 20,...Ch. 11.1 - Sketching Scalar Multiples In Exercises 25 and 26,...Ch. 11.1 - Sketching Scalar Multiples In Exercises 25 and 26,...Ch. 11.1 - Using Vector Operations In Exercises27 and 28,...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 - 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 - Sketching a Vector In Exercises 29-34, use the...Ch. 11.1 - Finding a Magnitude of a Vector In Exercises 2934,...Ch. 11.1 - Finding a Magnitude of a Vector In Exercises 2934,...Ch. 11.1 - Finding a Magnitude of a Vector In Exercises 2934,...Ch. 11.1 - Finding a Magnitude of a Vector InExercises 2934,...Ch. 11.1 - Finding a Magnitude of a Vector In Exercises 2934,...Ch. 11.1 - Finding a Magnitude of a Vector In Exercises 2934,...Ch. 11.1 - Finding a Unit Vector In Exercises 35-38, And the...Ch. 11.1 - Finding a Unit Vector In Exercises 35-38, And 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, And the...Ch. 11.1 - Finding Magnitudes In Exercises 39-42, find the...Ch. 11.1 - Finding Magnitudes In Exercises 39-42, find the...Ch. 11.1 - Finding Magnitudes In Exercises 39-42, find the...Ch. 11.1 - Finding Magnitudes In Exercises 39-42, find the...Ch. 11.1 - Using the Triangle Inequality In Exercises 43 and...Ch. 11.1 - Using the Triangle Inequality In Exercises 43 and...Ch. 11.1 - Finding a Vector In Exercises 45-48, find the...Ch. 11.1 - Finding a Vector In Exercises 45-48, find the...Ch. 11.1 - Finding a Vector In Exercises 45-48, find the...Ch. 11.1 - Finding a Vector In Exercises 45-48, find the...Ch. 11.1 - Finding a Vector In Exercises 49-52, find the...Ch. 11.1 - Finding a Vector In Exercises 49-52, find the...Ch. 11.1 - Finding a Vector In Exercises 49-52, find the...Ch. 11.1 - Finding a Vector In Exercises 49-52, 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 - Finding a Vector In Exercises 53-56, find the...Ch. 11.1 - Scalar and VectorIn your own words, state the...Ch. 11.1 - Scalar or VectorIdentify the quantity as a scalar...Ch. 11.1 - Using a Parallelogram Three vertices of a...Ch. 11.1 - HOW DO YOU SEE IT? Use the figure to determine...Ch. 11.1 - Finding Values In Exercises 6166, find aand bsuch...Ch. 11.1 - Finding Values In Exercises 6166, find aand bsuch...Ch. 11.1 - Finding Values In Exercises 6166, find aand bsuch...Ch. 11.1 - Finding Values In Exercises 6166, find aand bsuch...Ch. 11.1 - Finding Values In Exercises 6166, find aand bsuch...Ch. 11.1 - Finding Values In Exercises 6166, find aand bsuch...Ch. 11.1 - Finding Unit Vectors In Exercises 67-72, find a...Ch. 11.1 - Finding Unit Vectors In Exercises 67-72, find a...Ch. 11.1 - Finding Unit Vectors In Exercises 67-72, find a...Ch. 11.1 - Finding Unit Vectors In Exercises 67-72, find a...Ch. 11.1 - Finding Unit Vectors In Exercises 67-72, find a...Ch. 11.1 - Finding Unit Vectors In Exercises 67-72, find a...Ch. 11.1 - Finding a Vector In Exercises 73 and 74, find the...Ch. 11.1 - Finding a Vector In Exercises 73 and 74, find the...Ch. 11.1 - Resultant Force Forces with magnitudes of 500...Ch. 11.1 - Numerical and Graphical Analysis Forces with...Ch. 11.1 - Resultant Force Three forces with magnitudes of 75...Ch. 11.1 - Resultant Force Three forces with magnitudes of...Ch. 11.1 - Think About ItConsider two forces of equal...Ch. 11.1 - Cable TensionDetermine the tension in each cable...Ch. 11.1 - Projectile Motion A gun with a muzzle velocity...Ch. 11.1 - Shared Load To carry a 100-pound cylindrical...Ch. 11.1 - Navigation A plane is flying with a bearing of...Ch. 11.1 - Navigation A plane flies at a constant ground...Ch. 11.1 - True or False? In Exercises 85-94, determine...Ch. 11.1 - True or False? In Exercises 85-94, determine...Ch. 11.1 - True or False? In Exercises 85-94, determine...Ch. 11.1 - True or False? In Exercises 85-94, determine...Ch. 11.1 - True or False? In Exercises 85-94, determine...Ch. 11.1 - True or False? In Exercises 85-94, determine...Ch. 11.1 - Proof Prove that u=(cos)i(sin)jandv=(sin)i+(cos)j...Ch. 11.1 - Geometry Using vectors, prove that die line...Ch. 11.1 - GeometryUsing vectors, prove that the diagonals of...Ch. 11.1 - ProofProve that the vector w= ||u||v +...Ch. 11.1 - Using a Vector Consider the vector u=x,y. Describe...Ch. 11.1 - PUTNAM EXAM CHALLENGE A coast artillery gun can...Ch. 11.2 - CONCEPT CHECK Describing Coordinates A point in...Ch. 11.2 - Plotting Points In Exercises 5-8, plot the points...Ch. 11.2 - Plotting Points In Exercises 5-8, plot the points...Ch. 11.2 - Plotting Points In Exercises 5-8. plot the points...Ch. 11.2 - Plotting Points In Exercises 5-8. plot the points...Ch. 11.2 - Finding Coordinates of a Point In Exercises 9-12,...Ch. 11.2 - Finding Coordinates of a Point In Exercises 9-12,...Ch. 11.2 - Finding Coordinates of a Point In Exercises 9-12,...Ch. 11.2 - Finding Coordinates of a Point In Exercises 9-12,...Ch. 11.2 - Think About ItWhat is the z-coordinate of any...Ch. 11.2 - Think About It What is the x-coordinate of any...Ch. 11.2 - 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 - 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 - 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 - 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 - Finding the Distance Between Two Points in SpaceIn...Ch. 11.2 - Finding the Distance Between Two Points in Space...Ch. 11.2 - Finding the Distance Between Two Points in SpaceIn...Ch. 11.2 - Finding the Distance Between Two Points in Space...Ch. 11.2 - Classifying a Triangle In Exercises 29-32, find...Ch. 11.2 - Classifying a Triangle In Exercises 29-32, find...Ch. 11.2 - Classifying a Triangle In Exercises 29-32, find...Ch. 11.2 - Classifying a Triangle In Exercises 29-32, find...Ch. 11.2 - Think About It The triangle in Exercise 27 is...Ch. 11.2 - Think About It The triangle in Exercise 28 is...Ch. 11.2 - Finding the Midpoint In Exercises 33-36, find the...Ch. 11.2 - Finding the Midpoint In Exercises 33-36, find the...Ch. 11.2 - Finding the Midpoint In Exercises 33-36, find the...Ch. 11.2 - Finding the Midpoint In Exercises 33-36, find the...Ch. 11.2 - Finding the Equation of a Sphere In Exercises...Ch. 11.2 - Finding the Equation of a Sphere In Exercises...Ch. 11.2 - Finding the Equation of a Sphere In Exercises...Ch. 11.2 - Finding the Equation of a Sphere In Exercises...Ch. 11.2 - Finding the Equation of a Sphere In Exercises...Ch. 11.2 - Finding the Equation of a Sphere In Exercises...Ch. 11.2 - Finding the Equation of a Sphere In Exercises...Ch. 11.2 - Finding the Equation of a Sphere In Exercises...Ch. 11.2 - Finding the Component Form of a Vector in Space In...Ch. 11.2 - Finding the Component Form of a Vector in Space In...Ch. 11.2 - Finding the Component Form of a Vector in Space In...Ch. 11.2 - Finding the Component Form of a Vector in Space In...Ch. 11.2 - Writing a Vector in Different Forms In Exercises...Ch. 11.2 - Writing a Vector in Different Forms In Exercises...Ch. 11.2 - Finding the Component Form of a Vector in Space In...Ch. 11.2 - Finding the Component Form of a Vector in Space In...Ch. 11.2 - Finding a Terminal Point In Exercises 55 and 56,...Ch. 11.2 - Finding a Terminal Point In Exercises 55 and 56,...Ch. 11.2 - Finding Scalar Multiples In Exercises 57 and 58,...Ch. 11.2 - Finding Scalar Multiples In Exercises 57 and 58....Ch. 11.2 - Finding a Vector In Exercises 5760, find the...Ch. 11.2 - Finding a Vector In Exercises 59-62, find the...Ch. 11.2 - Finding a Vector In Exercises 5760, find the...Ch. 11.2 - Finding a Vector In Exercise 59-62, find the...Ch. 11.2 - Parallel Vectors In Exercises 63-66, determine...Ch. 11.2 - Parallel Vectors In Exercises 63-66, determine...Ch. 11.2 - Parallel Vectors In Exercises 63-66, determine...Ch. 11.2 - Parallel Vectors In Exercises 63-66, determine...Ch. 11.2 - Using Vectors to Determine Collinear Points In...Ch. 11.2 - Using Vectors to Determine Collinear Points In...Ch. 11.2 - Using Vectors to Determine Collinear Points In...Ch. 11.2 - Using Vectors to Determine Collinear Points In...Ch. 11.2 - Verifying a Parallelogram In Exercises 71 and 72,...Ch. 11.2 - Verifying a Parallelogram In Exercise 71 and 72,...Ch. 11.2 - Finding the Magnitude In Exercises 7176, find the...Ch. 11.2 - Finding the Magnitude In Exercises 7176, find the...Ch. 11.2 - Finding the Magnitude In Exercises 73-78, find the...Ch. 11.2 - Finding the Magnitude In Exercises 73-78, find the...Ch. 11.2 - Finding the Magnitude In Exercises 73-78, find the...Ch. 11.2 - Finding the Magnitude In Exercises 73-78, find the...Ch. 11.2 - Finding Unit Vectors In Exercises 79-82, find a...Ch. 11.2 - Finding Unit Vectors In Exercises 79-82, find a...Ch. 11.2 - Finding Unit Vectors In Exercises 79-82, find a...Ch. 11.2 - Finding Unit Vectors In Exercises 79-82, find a...Ch. 11.2 - Finding a Vector In Exercises 83-86, find the...Ch. 11.2 - Finding a Vector In Exercises 83-86, find the...Ch. 11.2 - Finding a Vector In Exercises 83-86, find the...Ch. 11.2 - Finding a Vector In Exercises 83-86, find the...Ch. 11.2 - Sketching a Vector In Exercises 87 and 88, sketch...Ch. 11.2 - Sketching a Vector In Exercises 87 und 88, sketch...Ch. 11.2 - Finding a Point Using Vectors In Exercises 89 and...Ch. 11.2 - Finding a Point Using Vectors In Exercises 89 and...Ch. 11.2 - EXPLORING CONCEPTS Writing The initial and...Ch. 11.2 - HOW DO YOU SEE IT? Determine (x, y, z) for each...Ch. 11.2 - Using Vectors Consider two nonzero vectors u and...Ch. 11.2 - Using Vectors Let u=i+j,v=j+k, and w=an+bv (a)...Ch. 11.2 - Diagonal of a Cube Find the component form of the...Ch. 11.2 - Tower Guy Wire The guy wire supporting a...Ch. 11.2 - Auditorium Lights The lights in an auditorium are...Ch. 11.2 - Think About It Suppose the length of each cable in...Ch. 11.2 - Load Supports Find the tension in each of the...Ch. 11.2 - Construction A precast concrete wall is...Ch. 11.2 - Geometry Write an equation whose graph consists of...Ch. 11.2 - Distance FormulaGive the formula for the distance...Ch. 11.2 - Standard Equation of a SphereGive the standard...Ch. 11.2 - Parallel Vectors State the definition of parallel...Ch. 11.2 - Using a Triangle and Vectors Let A, B, and Cbe...Ch. 11.2 - Using Vectors Let r=x,y,zandr0=1,1,1. Describe the...Ch. 11.3 - Find (a) u.v (b) u.u (c) (b) (c)(d) (e) v(f)...Ch. 11.3 - Find (a) u.v (b) u.u (c) (b) (c)(d) (e) v(f)...Ch. 11.3 - Find (a) u.v (b) u.u (c) (b) (c)(d) (e) v(f)...Ch. 11.3 - Finding Dot Products In Exercises 18, find (a) u...Ch. 11.3 - Finding Dot Products In Exercises 3-10, find (a)...Ch. 11.3 - Finding Dot Products In Exercises 18, find (a) u...Ch. 11.3 - Finding Dot Products In Exercises 3-10, find (a)...Ch. 11.3 - Finding Dot Products In Exercises 3-10, find (a)...Ch. 11.3 - Finding the Angle Between Two Vectors In Exercises...Ch. 11.3 - Finding the Angle Between Two Vectors In Exercises...Ch. 11.3 - Finding the Angle Between Two Vectors In Exercises...Ch. 11.3 - Finding the Angle Between Two Vectors In Exercises...Ch. 11.3 - Find the angle between the vectors (a) in radians...Ch. 11.3 - Finding the Angle Between Two Vectors In Exercises...Ch. 11.3 - Finding the Angle Between Two Vector In Exercises...Ch. 11.3 - Finding the Angle Between Two Vectors In Exercises...Ch. 11.3 - Use the alternative form of dot product to find...Ch. 11.3 - Use the alternative form of dot product to find...Ch. 11.3 - Determine whether u and v are orthogonal, parallel...Ch. 11.3 - Determine whether u and v are orthogonal, parallel...Ch. 11.3 - Determine whether u and v are orthogonal,...Ch. 11.3 - Determine whether u and v are orthogonal, parallel...Ch. 11.3 - Determine whether u and v are orthogonal, parallel...Ch. 11.3 - Determine whether u and v are orthogonal, parallel...Ch. 11.3 - The vertices of a triangle are given. Determine...Ch. 11.3 - The vertices of a triangle are given. Determine...Ch. 11.3 - The vertices of a triangle are given. Determine...Ch. 11.3 - Classifying a Triangle In Exercises 27-30, the...Ch. 11.3 - Finding Direction Angles In Exercises 31-36, find...Ch. 11.3 - Finding Direction Angles In Exercises 31-36 find...Ch. 11.3 - Finding Direction Angles In Exercises 2934, find...Ch. 11.3 - Find the direction cosines and angles of u and...Ch. 11.3 - Finding Direction Angles In Exercises 31-36, find...Ch. 11.3 - Finding Direction Angles In Exercises 31-36, find...Ch. 11.3 - (a) Find the projection of u onto v and (b) find...Ch. 11.3 - (a) Find the projection of u onto v and (b) find...Ch. 11.3 - (a) Find the projection of u onto v and (b) find...Ch. 11.3 - (a) Find the projection of u onto v and (b) find...Ch. 11.3 - (a) Find the projection of u onto v and (b) find...Ch. 11.3 - (a) Find the projection of u onto v and (b) find...Ch. 11.3 - Finding the Projection of u onto v In Exercises...Ch. 11.3 - Finding the Projection of u onto v In Exercises...Ch. 11.3 - Dot ProductDefine the dot product of vectors u and...Ch. 11.3 - Orthogonal Vectors Stale the definition of orthog...Ch. 11.3 - Using Vectors Determine which of the following are...Ch. 11.3 - Direction Cosines Describe direction cosines and...Ch. 11.3 - ProjectionGive a geometric description of the...Ch. 11.3 - Projection What can be said about the vectors u...Ch. 11.3 - When the projection of u onto v has the same...Ch. 11.3 - What is known about the angle between two...Ch. 11.3 - The vector u=3240,1450,2235 gives the number of...Ch. 11.3 - Repeat exercise 49 after decreasing the prices by...Ch. 11.3 - Orthogonal Vectors In Exercises 51-54, find two...Ch. 11.3 - Find two vectors in opposite directions that are...Ch. 11.3 - Find two vectors in opposite directions that are...Ch. 11.3 - Find two vectors in opposite directions that are...Ch. 11.3 - Find the angle between a cubes diagonal and one of...Ch. 11.3 - Find the angle between the diagonal of a cube and...Ch. 11.3 - A 48,000-pound truck is parked on a 100 slope (see...Ch. 11.3 - A 5400-pound sports utility vehicle is parked on...Ch. 11.3 - An object is pulled 10 feet across a floor using a...Ch. 11.3 - An wagon is pulled by exerting a force of 65...Ch. 11.3 - A car is towed using a force of 1600 newtons. The...Ch. 11.3 - Work A sled is pulled by exerting a force of 100...Ch. 11.3 - Determine whether the statement is true or false....Ch. 11.3 - Determine whether the statement is true or false....Ch. 11.3 - In exercise y=x2,y=x1/3 (a) Find all points of...Ch. 11.3 - In exercise y=x3,y=x1/3 (a) Find all points of...Ch. 11.3 - In exercise y=1x2,y=x21 (a) Find all points of...Ch. 11.3 - In exercise (y+1)2=x,y=x31 (a) Find all points of...Ch. 11.3 - Use vectors to prove that diagonals of a rhombus...Ch. 11.3 - Use vectors to prove that parallelogram is a...Ch. 11.3 - Consider a regular tetrahedron with vertices...Ch. 11.3 - Consider the vectors u=cos,sin,0 and v=cos,sin,0...Ch. 11.3 - Proof Prove thatu-v2=u2+v22uv.Ch. 11.3 - Prove the Cauchy-Schwarz inequality |u.v|uv.Ch. 11.3 - Prove the triangle inequality |u+v|u+v.Ch. 11.3 - Prove theorem 11.6.Ch. 11.4 - Cross Product of Unit Vectors In Exercises 3-6,...Ch. 11.4 - Cross Product of Unit Vectors In Exercises 16,...Ch. 11.4 - Cross Product of Unit Vectors In Exercises 3-6,...Ch. 11.4 - Cross Product of Unit Vectors In Exercises 16,...Ch. 11.4 - Cross Product of Unit Vectors In Exercises 3-6,...Ch. 11.4 - Cross Product of Unit Vectors In Exercises 3-6,...Ch. 11.4 - Finding Cross Products In Exercises 7-10, find (a)...Ch. 11.4 - Finding Cross Products In Exercises 7-10, find (a)...Ch. 11.4 - Finding Cross Products In Exercises 7-10, find (a)...Ch. 11.4 - Finding Cross Products In Exercises 710, find (a)...Ch. 11.4 - Finding a Cross Product In Exercises 1116, find u...Ch. 11.4 - Finding a Cross Product In Exercises 1116, find u...Ch. 11.4 - Finding a Cross Product In Exercises 1116, find u...Ch. 11.4 - Finding a Cross Product In Exercises 1116, find u...Ch. 11.4 - Finding a Cross Product In Exercises 11-14. find...Ch. 11.4 - Finding a Cross Product In Exercises 11-14, find...Ch. 11.4 - Finding a Unit Vector In Exercises 15-18. Find a...Ch. 11.4 - Finding a Unit Vector In Exercises 15-18. Find a...Ch. 11.4 - Finding a Unit Vector In Exercises 15-18. Find a...Ch. 11.4 - Finding a Unit Vector In Exercises 15-18. find a...Ch. 11.4 - Area In Exercises 19-22, find the area of the...Ch. 11.4 - Area In Exercises 19-22, find the area of the...Ch. 11.4 - Area In Exercises 19-22, find the area of the...Ch. 11.4 - Area In Exercises 19-22, find the area of the...Ch. 11.4 - Area In Exercises 23 and 24, verify that the...Ch. 11.4 - Area In Exercises 23 and 24, verify that the...Ch. 11.4 - Area In Exercises 25 and 26, find the area of the...Ch. 11.4 - Area In Exercises 25 and 26, find the area of the...Ch. 11.4 - Torque The brakes on a bicycle are applied using a...Ch. 11.4 - Torque Both the magnitude and the direction of the...Ch. 11.4 - Optimization A force of 180 pounds acts on the...Ch. 11.4 - Optimization A force of 56 pounds acts on the pipe...Ch. 11.4 - Finding a Triple Scalar Product In Exercises...Ch. 11.4 - Finding a Triple Scalar Product In Exercises...Ch. 11.4 - Finding a Triple Scalar Product In Exercises...Ch. 11.4 - Finding a Triple Scalar Product In Exercises...Ch. 11.4 - Volume In Exercises 35 and 36. use the triple...Ch. 11.4 - Volume In Exercises 35 and 36, use the triple...Ch. 11.4 - Volume In Exercises 37 and 38, find the volume of...Ch. 11.4 - Volume In Exercises 37 and 38, find the volume of...Ch. 11.4 - EXPLORING CONCEPTS Comparing Dot Products Identify...Ch. 11.4 - EXPLORING CONCEPTS Using Dot and Cross Products...Ch. 11.4 - WRITING ABOUT CONCEPTS Cross Product Define the...Ch. 11.4 - WRITING ABOUT CONCEPTS Cross Product State the...Ch. 11.4 - WRITING ABOUT CONCEPTS Magnitude When the...Ch. 11.4 - HOW DO YOU SEE IT? The vertices of a triangle in...Ch. 11.4 - True or False? In Exercises 45-46, determine...Ch. 11.4 - True or False? In Exercises 4750. determine...Ch. 11.4 - True or False? In Exercises 43-46, determine...Ch. 11.4 - True or False? In Exercises 43-46, determine...Ch. 11.4 - Proof In Exercises 47-52, prove the property of...Ch. 11.4 - Proof In Exercises 47-52. prove the property of...Ch. 11.4 - Proof In Exercises 47-52, prove the property of...Ch. 11.4 - Proof In Exercises 47-52. prove the property of...Ch. 11.4 - Proof In Exercises 47-52, prove the property of...Ch. 11.4 - Proof In Exercises 47-52. prove the property of...Ch. 11.4 - Proof Prove that uv=uv if u and v are orthogonal.Ch. 11.4 - Proof Prove that u(vw)=(uw)v(uv)wCh. 11.4 - Proof Prove Theorem 11.9.Ch. 11.5 - CONCEPT CHECK Parametric and Symmetric Equations...Ch. 11.5 - Checking Points on a Line In Exercises 5 and 6,...Ch. 11.5 - Checking Points on a Line In Exercises 5 and 6,...Ch. 11.5 - Finding Parametric and Symmetric Equations In...Ch. 11.5 - Finding Parametric and Symmetric Equations In...Ch. 11.5 - Finding Parametric and Symmetric Equations In...Ch. 11.5 - Finding Parametric and Symmetric Equations In...Ch. 11.5 - Finding Parametric and Symmetric Equations In...Ch. 11.5 - Finding Parametric and Symmetric Equations In...Ch. 11.5 - Finding Parametric and Symmetric Equations In...Ch. 11.5 - Finding Parametric and Symmetric Equations In...Ch. 11.5 - Finding Parametric and Symmetric Equations In...Ch. 11.5 - Finding Parametric and Symmetric Equations In...Ch. 11.5 - Finding Parametric Equations In Exercises 17-24....Ch. 11.5 - Finding Parametric Equations In Exercises 17-24....Ch. 11.5 - Finding Parametric Equations In Exercises 17-24....Ch. 11.5 - Finding Parametric Equations In Exercises 17-24....Ch. 11.5 - Finding Parametric Equations In Exercises 17-24,...Ch. 11.5 - Finding Parametric Equations In Exercises 17-24....Ch. 11.5 - Finding Parametric Equations In Exercises 17-24,...Ch. 11.5 - Finding Parametric Equations In Exercises 17-24....Ch. 11.5 - Using Parametric and Symmetric Equations In...Ch. 11.5 - Using Parametric and Symmetric Equations In...Ch. 11.5 - Using Parametric and Symmetric Equations In...Ch. 11.5 - Using Parametric and Symmetric Equations In...Ch. 11.5 - Determining Parallel Lines In Exercises 2528,...Ch. 11.5 - Determining Parallel Lines In Exercises 2528,...Ch. 11.5 - Determining Parallel Lines In Exercises 2528,...Ch. 11.5 - Determining Parallel Lines In Exercises 2528,...Ch. 11.5 - Finding a Point of Intersection In Exercises...Ch. 11.5 - Finding a Point of Intersection In Exercises...Ch. 11.5 - Finding a Point of Intersection In Exercises...Ch. 11.5 - Finding a Point of Intersection In Exercises...Ch. 11.5 - Checking Points in a Plane In Exercises 37 and 38,...Ch. 11.5 - Checking Points in a Plane In Exercises 37 and 38,...Ch. 11.5 - Finding an Equation of a Plane In Exercises39-44,...Ch. 11.5 - Finding an Equation of a Plane In Exercises39-44,...Ch. 11.5 - Finding an Equation of a Plane In Exercises39-44,...Ch. 11.5 - Finding an Equation of a Plane In Exercises39-44,...Ch. 11.5 - Finding an Equation of a Plane In Exercises39-44,...Ch. 11.5 - Finding an Equation of a Plane In Exercises39-44,...Ch. 11.5 - Finding an Equation of a Plane In Exercises45-56,...Ch. 11.5 - Finding an Equation of a Plane In Exercises45-56,...Ch. 11.5 - Finding an Equation of a Plane In Exercises45-56,...Ch. 11.5 - Finding an Equation of a Plane In Exercises45-56,...Ch. 11.5 - Finding an Equation of a Plane In Exercises45-56,...Ch. 11.5 - Finding an Equation of a Plane In Exercises45-56....Ch. 11.5 - Finding an Equation of a Plane In Exercises45-56,...Ch. 11.5 - Finding an Equation of a Plane In Exercises45-56,...Ch. 11.5 - Finding an Equation of a Plane In Exercises45-56,...Ch. 11.5 - Finding an Equation of a Plane In Exercises45-56,...Ch. 11.5 - Finding an Equation of a Plane In Exercises45-56,...Ch. 11.5 - Finding an Equation of a Plane In Exercises45-56,...Ch. 11.5 - Finding an Equation of a Plane In Exercises 57-60,...Ch. 11.5 - Finding an Equation of a Plane In Exercises 57-60,...Ch. 11.5 - Finding an Equation of a Plane In Exercises 57-60,...Ch. 11.5 - Finding an Equation of a Plane In Exercises 57-60,...Ch. 11.5 - Intersection of Planes In Exercises 65-68, (a)...Ch. 11.5 - Intersection of Planes In Exercises 65-68, (a)...Ch. 11.5 - Comparing Planes In Exercises 69-74, determine...Ch. 11.5 - Comparing Planes In Exercises 69-74, determine...Ch. 11.5 - Comparing Planes In Exercises 69-74, determine...Ch. 11.5 - Comparing Planes In Exercises 69-74, determine...Ch. 11.5 - Comparing Planes In Exercises 69-74, determine...Ch. 11.5 - Comparing Planes In Exercises 69-74, determine...Ch. 11.5 - Sketching a Graph of a Plane In Exercises75-82,...Ch. 11.5 - Sketching a Graph of a Plane In Exercises75-82,...Ch. 11.5 - Sketching a Graph of a Plane In Exercises75-82,...Ch. 11.5 - Sketching a Graph of a Plane In Exercises75-82,...Ch. 11.5 - Sketching a Graph of a Plane In Exercises75-82,...Ch. 11.5 - Sketching a Graph of a Plane In Exercises75-82,...Ch. 11.5 - Sketching a Graph of a Plane In Exercises 6370,...Ch. 11.5 - Sketching a Graph of a Plane In Exercises 6370,...Ch. 11.5 - Parallel Planes In Exercises 7174, determine...Ch. 11.5 - Parallel Planes In Exercises 7174, determine...Ch. 11.5 - Parallel Planes In Exercises 7174, determine...Ch. 11.5 - Parallel Planes In Exercises 7174, determine...Ch. 11.5 - Intersection of a Plane and a Line In Exercises...Ch. 11.5 - Intersection of a Plane and a Line In Exercises...Ch. 11.5 - Intersection of a Plane and a Line In Exercises...Ch. 11.5 - Intersection of a Plane and a Line In Exercises...Ch. 11.5 - Finding the Distance Between a Point and a Plane...Ch. 11.5 - Finding the Distance Between a Point and a Plane...Ch. 11.5 - Finding the Distance Between a Point and a Plane...Ch. 11.5 - Finding the Distance Between a Point and a Plane...Ch. 11.5 - Finding the Distance Between Two Parallel Planes...Ch. 11.5 - Finding the Distance Between Two Parallel Planes...Ch. 11.5 - Finding the Distance Between Two Parallel Planes...Ch. 11.5 - Finding the Distance Between Two Parallel Planes...Ch. 11.5 - Finding the Distance Between a Point and a Line In...Ch. 11.5 - Finding the Distance Between a Point and a Line In...Ch. 11.5 - Finding the Distance Between a Point and a Line In...Ch. 11.5 - Finding the Distance Between a Point and a Line In...Ch. 11.5 - Finding the Distance Between Two Parallel Lines In...Ch. 11.5 - Finding the Distance Between Two Parallel Lines In...Ch. 11.5 - Standard Equation of a Plane in Space Give the...Ch. 11.5 - Intersection of Two PlanesDescribe a method of...Ch. 11.5 - Parallel and Perpendicular PlanesDescribe a method...Ch. 11.5 - Normal Vector Let L1 and L2 be nonparallel lines...Ch. 11.5 - HOW DO YOU SEE IT? Match the general equation with...Ch. 11.5 - Modeling DataPersonal consumption expenditures (in...Ch. 11.5 - Mechanical Design The figure shows a chute at the...Ch. 11.5 - Distance Two insects are crawling along different...Ch. 11.5 - Finding an Equation of a Sphere Find the standard...Ch. 11.5 - Finding a Point of Intersection Find the point of...Ch. 11.5 - Finding the Distance Between a Plane and a Line...Ch. 11.5 - Finding a Point of IntersectionFind the point of...Ch. 11.5 - Finding Parametric Equations Find a set of...Ch. 11.5 - True or False? In Exercises 113-118, determine...Ch. 11.5 - True or False? In Exercises 109114, determine...Ch. 11.5 - True or False? In Exercises 113-118, determine...Ch. 11.5 - True or False? In Exercises 113-118, determine...Ch. 11.5 - True or False? In Exercises 109114, determine...Ch. 11.5 - True or False? In Exercises 113-118, determine...Ch. 11.6 - CONCEPT CHECK Classifying an Equation What does...Ch. 11.6 - Matching In Exercises 5-10, match the equation...Ch. 11.6 - Matching In Exercises 5-10, match the equation...Ch. 11.6 - Matching In Exercises 5-10, match the equation...Ch. 11.6 - Matching In Exercises 5-10, match the equation...Ch. 11.6 - Matching In Exercises 5-10, match the equation...Ch. 11.6 - Matching In Exercises 5-10, match the equation...Ch. 11.6 - Sketching a Surface in Space In Exercises 712,...Ch. 11.6 - Sketching a Surface in Space In Exercises 712,...Ch. 11.6 - Sketching a Surface in Space In Exercises11-14,...Ch. 11.6 - Sketching a Surface in Space In Exercises11-14,...Ch. 11.6 - Sketching a Surface in Space In Exercises11-14,...Ch. 11.6 - Sketching a Surface in Space In Exercises 712,...Ch. 11.6 - Sketching a Quadric Surface In Exercises15-26,...Ch. 11.6 - Sketching a Quadric Surface In Exercises15-26,...Ch. 11.6 - Sketching a Quadric Surface In Exercises15-26,...Ch. 11.6 - Sketching a Quadric Surface In Exercises15-26,...Ch. 11.6 - Sketching a Quadric Surface In Exercises15-26,...Ch. 11.6 - Sketching a Quadric Surface In Exercises15-26,...Ch. 11.6 - Sketching a Quadric Surface In Exercises15-26,...Ch. 11.6 - Sketching a Quadric Surface In Exercises15-26,...Ch. 11.6 - Sketching a Quadric Surface In Exercises15-26,...Ch. 11.6 - Sketching a Quadric Surface In Exercises15-26,...Ch. 11.6 - Sketching a Quadric Surface In Exercises15-26,...Ch. 11.6 - Sketching a Quadric Surface In Exercises15-26,...Ch. 11.6 - Cylinder State the definition of a cylinder.Ch. 11.6 - Trace of a Surface What is meant by the trace...Ch. 11.6 - Quadric SurfacesIdentify the six quadric surfaces...Ch. 11.6 - Classifying an Equation What does the equation 4X2...Ch. 11.6 - HOW DO YOU SEE IT? 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In Exercises 4952, determine...Ch. 11.6 - True or False? In Exercises 4952, determine...Ch. 11.6 - True or False? In Exercises 4952, determine...Ch. 11.6 - True or False? In Exercises 4952, determine...Ch. 11.7 - Cylindrical-to-Rectangular Conversion In Exercises...Ch. 11.7 - Cylindrical-to-Rectangular Conversion In Exercises...Ch. 11.7 - Cylindrical-to-Rectangular Conversion In Exercises...Ch. 11.7 - Cylindrical-to-Rectangular Conversion In Exercises...Ch. 11.7 - Cylindrical-to-Rectangular Conversion In Exercises...Ch. 11.7 - Cylindrical-to-Rectangular Conversion In Exercises...Ch. 11.7 - Rectangular-to-Cylindrical Conversion In Exercises...Ch. 11.7 - Rectangular-to-Cylindrical Conversion In Exercises...Ch. 11.7 - Rectangular-to-Cylindrical Conversion In Exercises...Ch. 11.7 - Rectangular-to-Cylindrical Conversion In Exercises...Ch. 11.7 - Rectangular-to-Cylindrical Conversion In Exercises...Ch. 11.7 - Rectangular-to-Cylindrical Conversion In Exercises...Ch. 11.7 - Rectangular-to-Cylindrical Conversion In Exercises...Ch. 11.7 - Rectangular-to-Cylindrical Conversion In Exercises...Ch. 11.7 - Rectangular-to-Cylindrical Conversion In Exercises...Ch. 11.7 - Rectangular-to-Cylindrical Conversion In Exercises...Ch. 11.7 - Rectangular-to-Cylindrical Conversion In Exercises...Ch. 11.7 - Rectangular-to-Cylindrical Conversion In Exercises...Ch. 11.7 - Rectangular-to-Cylindrical Conversion In Exercises...Ch. 11.7 - Rectangular-to-Cylindrical Conversion In Exercises...Ch. 11.7 - Cylindrical-to-Rectangular Conversion In Exercises...Ch. 11.7 - Cylindrical-to-Rectangular Conversion In Exercises...Ch. 11.7 - Cylindrical-to-Rectangular Conversion In Exercises...Ch. 11.7 - Cylindrical-to-Rectangular Conversion In Exercises...Ch. 11.7 - Cylindrical-to-Rectangular Conversion In Exercises...Ch. 11.7 - Cylindrical-to-Rectangular Conversion In Exercises...Ch. 11.7 - Cylindrical-to-Rectangular Conversion In Exercises...Ch. 11.7 - Cylindrical-to-Rectangular Conversion In Exercises...Ch. 11.7 - Rectangular-to-Spherical Conversion In...Ch. 11.7 - Rectangular-to-Spherical Conversion In...Ch. 11.7 - Rectangular-to-Spherical Conversion In...Ch. 11.7 - Rectangular-to-Spherical Conversion In Exercises...Ch. 11.7 - Rectangular-to-Spherical Conversion In...Ch. 11.7 - Rectangular-to-Spherical Conversion In...Ch. 11.7 - Spherical-to-Rectangular Conversion In...Ch. 11.7 - Spherical-to-Rectangular Conversion In...Ch. 11.7 - Spherical-to-Rectangular Conversion In...Ch. 11.7 - Spherical-to-Rectangular Conversion In...Ch. 11.7 - Spherical-to-Rectangular Conversion In Exercises...Ch. 11.7 - Spherical-to-Rectangular Conversion In Exercises...Ch. 11.7 - Rectangular-to-Spherical Conversion In Exercises...Ch. 11.7 - Rectangular-to-Spherical Conversion In Exercises...Ch. 11.7 - Rectangular-to-Spherical Conversion In Exercises...Ch. 11.7 - Rectangular-to-Spherical Conversion In Exercises...Ch. 11.7 - Rectangular-to-Spherical Conversion In Exercises...Ch. 11.7 - Rectangular-to-Spherical Conversion In Exercises...Ch. 11.7 - Rectangular-to-Spherical Conversion In Exercises...Ch. 11.7 - Rectangular-to-Spherical Conversion In Exercises...Ch. 11.7 - Spherical-to-Rectangular Conversion InExercises...Ch. 11.7 - Spherical-to-Rectangular Conversion In Exercises...Ch. 11.7 - Spherical-to-Rectangular Conversion In Exercises...Ch. 11.7 - Spherical-to-Rectangular Conversion In Exercises...Ch. 11.7 - Spherical-to-Rectangular Conversion In Exercises...Ch. 11.7 - Spherical-to-Rectangular Conversion In Exercises...Ch. 11.7 - Spherical-to-Rectangular Conversion In Exercises...Ch. 11.7 - Spherical-to-Rectangular Conversion In Exercises...Ch. 11.7 - Cylindrical-to-Spherical Conversion In Exercises...Ch. 11.7 - Cylindrical-to-Spherical Conversion In Exercises...Ch. 11.7 - Cylindrical-to-Spherical Conversion In Exercises...Ch. 11.7 - Cylindrical-to-Spherical Conversion In Exercises...Ch. 11.7 - Cylindrical-to-Spherical Conversion In Exercises...Ch. 11.7 - Spherical-to-Cylindrical Conversion In Exercises...Ch. 11.7 - Spherical-to-Cylindrical Conversion In Exercises...Ch. 11.7 - Spherical-to-Cylindrical Conversion In Exercises...Ch. 11.7 - Spherical-to-Cylindrical Conversion In Exercises...Ch. 11.7 - Spherical-to-Cylindrical Conversion In Exercises...Ch. 11.7 - Spherical-to-Cylindrical Conversion In Exercises...Ch. 11.7 - Matching In Exercises 71-76, match the equation...Ch. 11.7 - Matching In Exercises 71-76, match the equation...Ch. 11.7 - Matching In Exercises 71-76, match the equation...Ch. 11.7 - Matching In Exercises 71-76, match the equation...Ch. 11.7 - Matching In Exercises 71-76, match the equation...Ch. 11.7 - Matching In Exercises 71-76, match the equation...Ch. 11.7 - Spherical Coordinates Explain why in spherical...Ch. 11.7 - HOW DO YOU SEE IT? 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Using Vectors to Determine Collinear Points In...Ch. 11 - Finding a Unit Vector Find a unit vector in the...Ch. 11 - Finding a Vector Find the vector v of magnitude 8...Ch. 11 - Finding Dot Products In Exercises 21 and 22, let...Ch. 11 - Finding Dot Products In Exercises 21 and 22, let...Ch. 11 - Finding the Angle Between Two Vectors In...Ch. 11 - Finding the Angle Between Two Vectors In Exercises...Ch. 11 - Finding the Angle Between Two Vectors In Exercises...Ch. 11 - Finding the Angle Between Two Vectors In...Ch. 11 - Comparing Vectors In Exercises 25 and 26,...Ch. 11 - Comparing VectorsIn Exercises 25 and 26, determine...Ch. 11 - Finding the Projection of u onto v InExercises...Ch. 11 - Finding the Projection of u onto v In Exercises 27...Ch. 11 - Finding the Projection of u onto v In Exercises 27...Ch. 11 - Finding the Projection of u onto v InExercises...Ch. 11 - Orthogonal Vectors Find two vectors in opposite...Ch. 11 - Work An object is pulled 8 feet across a floor...Ch. 11 - Finding Cross Products In Exercises 31 and 32,...Ch. 11 - Finding Cross Products In Exercises 3336, find (a)...Ch. 11 - Finding Cross Products In Exercises 3336, find (a)...Ch. 11 - Finding Cross Products In Exercises 31 and 32,...Ch. 11 - Finding a Unit Vector Find a unit vector that is...Ch. 11 - Area Find the area of the parallelogram that has...Ch. 11 - Torque The specifications for a tractor state that...Ch. 11 - Volume Use the triple scalar product to find the...Ch. 11 - Finding Parametric and Symmetric Equations In...Ch. 11 - Finding Parametric and Symmetric Equations In...Ch. 11 - Finding Parametric Equations In Exercises 4346,...Ch. 11 - Finding Parametric Equations In Exercises 39 and...Ch. 11 - Finding Parametric Equations InExercises 4346,...Ch. 11 - Finding Parametric Equations In Exercises 4346,...Ch. 11 - Finding an Equation of a Plane In Exercises 41-44,...Ch. 11 - Finding an Equation of a Plane In Exercises 41-44,...Ch. 11 - Finding an Equation of a Plane In Exercises 41-44,...Ch. 11 - Finding an Equation of a Plane In Exercises 41-44,...Ch. 11 - Distance Find the distance between the point (1,...Ch. 11 - Distance Find the distance between the point...Ch. 11 - Distance Find the distance between the planes...Ch. 11 - Distance Find the distance between the point...Ch. 11 - Sketching a Surface in Space In Exercises 49-58,...Ch. 11 - Sketching a Surface in Space In Exercises 49-58,...Ch. 11 - Sketching a Surface in Space In Exercises 49-58,...Ch. 11 - Sketching a Surface in Space In Exercises 49-58,...Ch. 11 - Sketching a Surface in Space In Exercises 49-58,...Ch. 11 - Sketching a Surface in Space In Exercises 49-58,...Ch. 11 - Sketching a Surface in Space In Exercises 49-58,...Ch. 11 - Sketching a Surface in Space In Exercises 49-58,...Ch. 11 - Sketching a Surface in Space In Exercises 49-58,...Ch. 11 - Sketching a Surface in Space In Exercises 49-58,...Ch. 11 - Surface of Revolution Find an equation for the...Ch. 11 - Surface of Revolution Find an equation for the...Ch. 11 - Converting Rectangular Coordinates In Exercises 67...Ch. 11 - Converting Rectangular Coordinates In Exercises 67...Ch. 11 - Cylindrical-to-Spherical Conversion In Exercises...Ch. 11 - Cylindrical-to-Spherical Conversion In Exercises...Ch. 11 - Spherical-to-Cylindrical Conversion In Exercises...Ch. 11 - Spherical-to-Cylindrical Conversion In Exercises...Ch. 11 - Converting a Rectangular Equation In Exercises 67...Ch. 11 - Converting a Rectangular Equation In Exercises 67...Ch. 11 - Cylindrical-to-Rectangular Conversion In Exercises...Ch. 11 - Cylindrical-to-Rectangular Conversion In Exercises...Ch. 11 - Spherical-to-Rectangular Conversion In Exercises...Ch. 11 - Spherical-to-Rectangular Conversion In Exercises...Ch. 11 - Proof Using vectors, prove the Law of Sines: If a,...Ch. 11 - Using an Equation Consider the function...Ch. 11 - Proof Using vectors, prove that the line segments...Ch. 11 - Proof Using vectors, prove that the diagonals of a...Ch. 11 - Distance (a) Find the shortest distance between...Ch. 11 - Orthogonal Vectors Let P0 be a point in the plane...Ch. 11 - 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