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7th Edition

Ron Larson + 1 other

Publisher: Cengage Learning

ISBN: 9781337552516

Chapter 11, Problem 25RE

Textbook Problem

Comparing Vectors In Exercises 25 and 26, determine whether u and v are orthogonal, parallel, or neither.

Calculus: Early Transcendental Functions

Ch. 11.1 - CONCEPT CHECK Scalar and Vector Describe the...Ch. 11.1 - CONCEPT CHECK Vector Two points and a vector are...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 - Finding a Magnitude of a Vector In Exercises...Ch. 11.1 - Finding a Magnitude of a Vector In Exercises...Ch. 11.1 - Finding a Magnitude of a Vector In Exercises...Ch. 11.1 - Finding a Magnitude of a Vector In Exercises...Ch. 11.1 - Finding a Magnitude of a Vector In Exercises...Ch. 11.1 - Finding a Magnitude of a Vector In Exercises...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 - 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 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 - EXPLORING CONCEPTS Think About It In Exercises 57...Ch. 11.1 - EXPLORING CONCEPTS Think About It In Exercises 57...Ch. 11.1 - EXPLORING CONCEPTS Think About It In Exercises 57...Ch. 11.1 - HOW DO YOU SEE IT? Use the figure to determine...Ch. 11.1 - Finding Values In Exercises 61-66, find a and b...Ch. 11.1 - Finding Values In Exercises 61-66, find a and b...Ch. 11.1 - Finding Values In Exercises 61-66, find a and b...Ch. 11.1 - Finding Values In Exercises 61-66, find a and b...Ch. 11.1 - Finding Values In Exercises 61-66, find a and b...Ch. 11.1 - Finding Values In Exercises 61-66, find a and b...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 - Cable Tension In Exercises 79 and 80, determine...Ch. 11.1 - Cable Tension In Exercises 79 and 79, determine...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 - 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 - Geometry Using vectors, prove tint the diagonals...Ch. 11.1 - Proof Prove that the vector w=uv+vu bisects the...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 - Coordinates in Space What is the y-coordinate of...Ch. 11.2 - CONCEPT CHECK Comparing Graphs Describe the graph...Ch. 11.2 - Parallel Vectors Explain how to determine whether...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 - 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 Space...Ch. 11.2 - Finding the Distance Between Two Points in Space...Ch. 11.2 - Finding the Distance Between Two Points in Space...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 - 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 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 - 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 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 59-62, find the...Ch. 11.2 - Finding a Vector In Exercises 59-62, find the...Ch. 11.2 - Finding a Vector In Exercises 59-62, 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 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 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 - Writing Lei r=(x,y,x) and r0=(1,1,1) Describe the...Ch. 11.2 - Writing Let r=x,y,z. Describe the set of all...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. 11.3 - Dot product What can you say about the relative...Ch. 11.3 - Direction Cosines Consider the vector v=v1,v2,v3....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 - Find (a) u.v (b) u.u (c) (b) (c)(d) (e) v(f)...Ch. 11.3 - Finding Dot Products In Exercises 3-10, find (a)...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 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 31-36, 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 - (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 - Explain why u + v.w is not defined, where u, v and...Ch. 11.3 - What can be said about the vectors u and v when...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 - A pallet truck is pulled by exerting a force of...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 - Prove the triangle inequality |u+v|u+v.Ch. 11.3 - Prove theorem 11.6.Ch. 11.3 - Prove the Cauchy-Schwarz inequality |u.v|uv.Ch. 11.4 - CONCEPT CHECK Vectors Explain what uv represents...Ch. 11.4 - CONCEPT CHECK Area Explain how to find the area of...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 - 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 7-10, find (a)...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 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 - EXPLORING CONCEPTS Cross Product Two nonzero...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 43-46, 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 - CONCEPT CHECK Normal Vector The equation of a...Ch. 11.5 - CONCEPT CHECK Plane Write an equation of a plane...Ch. 11.5 - CONCEPT CHECK Parallel Planes Explain how to find...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 29-32,...Ch. 11.5 - Determining Parallel Lines In Exercises 29-32,...Ch. 11.5 - Determining Parallel Lines In Exercises 29-32,...Ch. 11.5 - Determining Parallel Lines In Exercises 29-32,...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 - Parallel Planes In Exercises 61-64, determine...Ch. 11.5 - Parallel Planes In Exercises 61-64, determine...Ch. 11.5 - Parallel Planes In Exercises 61-64, determine...Ch. 11.5 - Parallel Planes In Exercises 61-64, determine...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 - 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 Exercises75-82,...Ch. 11.5 - Sketching a Graph of a Plane In Exercises75-82,...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 - EXPLORING CONCEPTS Planes Consider a line and a...Ch. 11.5 - EXPLORING CONCEPTS Planes How many planes are...Ch. 11.5 - EXPLORING CONCEPTS Think About It Do two distinct...Ch. 11.5 - HOW DO YOU SEE IT? Match the general equation with...Ch. 11.5 - Modeling Data Personal consumption expenditures...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 Parametric Equations Find a set 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 113-118, 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 113-118, determine...Ch. 11.5 - True or False? In Exercises 113-118, determine...Ch. 11.6 - CONCEPT CHECK, Quadric Surfaces How are quadric...Ch. 11.6 - CONCEPT CHECK Classifying an Equation What does...Ch. 11.6 - CONCEPT CHECK Trace of a Surface What is meant by...Ch. 11.6 - CONCEPT CHECK Think About It Does every...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 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 Exercises11-14,...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 - EXPLORING CONCEPTS Hyperboloid Explain how to...Ch. 11.6 - EXPLORING CONCEPTS Ellipsoid Is every trace of an...Ch. 11.6 - EXPLORING CONCEPTS Quadric Surface Is there a...Ch. 11.6 - HOW DO YOU SEE IT? 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Geography Because of the forces caused by its...Ch. 11.6 - Machine Design The top of a rubber bushing...Ch. 11.6 - Using a Hyperbolic Paraboloid Determine the...Ch. 11.6 - Intersection of Surfaces Explain why the curve of...Ch. 11.6 - Think About It Three types of classic topological...Ch. 11.7 - CONCEPT CHECK Cylindrical Coordinates Describe the...Ch. 11.7 - CONCEPT CHECK Spherical Coordinates Describe the...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 - 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Identify the surface graphed...Ch. 11.7 - Converting a Rectangular Equation In...Ch. 11.7 - Converting a Rectangular Equation In...Ch. 11.7 - Converting a Rectangular Equation In...Ch. 11.7 - Converting a Rectangular Equation In...Ch. 11.7 - Converting a Rectangular Equation In...Ch. 11.7 - Converting a Rectangular Equation In...Ch. 11.7 - Converting a Rectangular Equation In...Ch. 11.7 - Converting a Rectangular Equation In...Ch. 11.7 - Sketching a Solid In Exercises 87-90, sketch the...Ch. 11.7 - Sketching a Solid In Exercises 87-90, sketch the...Ch. 11.7 - Sketching a Solid In Exercises 87-90, sketch the...Ch. 11.7 - Sketching a Solid In Exercises 87-90, sketch the...Ch. 11.7 - Sketching a Solid In Exercises 91-94, sketch the...Ch. 11.7 - Sketching a Solid In Exercises 91-94, sketch the...Ch. 11.7 - Sketching a Solid In Exercises 91-94, sketch the...Ch. 11.7 - Sketching a Solid In Exercises 91-94, sketch the...Ch. 11.7 - EXPLORING CONCEPTS Think About It In Exercises...Ch. 11.7 - EXPLORING CONCEPTS Think About It In Exercises...Ch. 11.7 - EXPLORING CONCEPTS Think About It In Exercises...Ch. 11.7 - EXPLORING CONCEPTS Think About It In Exercises...Ch. 11.7 - EXPLORING CONCEPTS Think About It In Exercises...Ch. 11.7 - EXPLORING CONCEPTS Think About It In Exercises...Ch. 11.7 - True or False? 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In Exercises 101 and 102, determine...Ch. 11.7 - Intersection of Surfaces Identify the curve of...Ch. 11.7 - Intersection of Surfaces Identify the curve of...Ch. 11 - Writing Vectors in Different Forms In Exercises 1...Ch. 11 - Writing Vectors in Different Forms In Exercises 1...Ch. 11 - Finding a Vector In Exercises 3 and 4, find the...Ch. 11 - Finding a Vector In Exercises 3 and 4, find the...Ch. 11 - Finding Coordinates of a Point Find the...Ch. 11 - Using the Three-Dimensional Coordinate System...Ch. 11 - Finding the Distance Between Two Points in Space...Ch. 11 - Finding the Distance Between Two Points in Space...Ch. 11 - Finding the Equation of a Sphere In Exercises 9...Ch. 11 - Finding the Equation of a Sphere In Exercises 9...Ch. 11 - Finding the Equation of a Sphere In Exercises 11...Ch. 11 - Finding the Equation of a Sphere In Exercises 11...Ch. 11 - Writing a Vector in Different Forms In Exercises...Ch. 11 - Writing a Vector in Different Forms In Exercises...Ch. 11 - Finding a Vector In Exercises 15 and 16, find the...Ch. 11 - Finding a Vector In Exercises 15 and 16, find the...Ch. 11 - Using Vectors to Determine Collinear Points In...Ch. 11 - 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...Ch. 11 - Comparing Vectors In Exercises 25 and 26,...Ch. 11 - Comparing Vectors In Exercises 25 and 26,...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 - 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 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 A vertical force of 40 pounds acts on a...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 39 and...Ch. 11 - Finding Parametric Equations In Exercises 39 and...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 61...Ch. 11 - Converting Rectangular Coordinates In Exercises 61...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 - 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 - Volume (a) Find the volume of the solid bounded...Ch. 11 - Volume (a) Use the disk method to Find the volume...Ch. 11 - Proof Prove the following property of the cross...Ch. 11 - Using Parametric Equations Consider the line given...Ch. 11 - Sketching Graphs Sketch the graph of each equation...Ch. 11 - Sketching Graphs Sketch the graph of each equation...Ch. 11 - 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