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6th Edition

Ron Larson + 1 other

Publisher: Cengage Learning

ISBN: 9781285774770

Chapter 11.4, Problem 52E

Textbook Problem

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Proof In Exercises 47-52. prove the property of the cross product.

To determine

**To Prove:** The property of cross product

**Given:**

The cross-product property is;

**Proof:**

It has to prove

Let

And,

The vector

Calculus: Early Transcendental Functions (MindTap Course List)

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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? The four figures below are...Ch. 11.6 - Finding an Equation for a Surface of Revolution In...Ch. 11.6 - Finding an Equation of a Surface of Revolution In...Ch. 11.6 - Finding an Equation of a Surface of Revolution In...Ch. 11.6 - Finding an Equation of a Surface of Revolution In...Ch. 11.6 - Finding an Equation for a Surface of Revolution In...Ch. 11.6 - Finding an Equation for a Surface of Revolution In...Ch. 11.6 - Finding a Generating Curve In Exercises37-40, find...Ch. 11.6 - Finding a Generating Curve In Exercises37-40, find...Ch. 11.6 - Finding the Volume of a Solid In Exercises 41 and...Ch. 11.6 - Finding the Volume of a Solid In Exercises 41 and...Ch. 11.6 - Analyzing a Trace In Exercises 43 and 44, analyze...Ch. 11.6 - Analyzing a Trace In Exercises 43 and 44, analyze...Ch. 11.6 - Finding an Equation of a Surface In Exercises 45...Ch. 11.6 - Finding an Equation of a Surface In Exercises 45...Ch. 11.6 - 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. 11.6 - 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. 11.6 - 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 - Volume (a) Find the volume of the solid bounded...Ch. 11 - Volume (a) Use the disk method to Find the volume...Ch. 11 - ProofProve 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 - Tetherball A tetherball weighing 1 pound is pulled...Ch. 11 - Towing A loaded barge is being towed by two...Ch. 11 - Proof Consider the vectors...Ch. 11 - Latitude-Longitude System Los Angeles is located...Ch. 11 - Distance Between a Point and a Plane Consider the...Ch. 11 - Distance Between Parallel Planes Show that the...Ch. 11 - Intersection of Planes Show that the curve of...Ch. 11 - Vector AlgebraRead the article Tooth Tables:...

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