Explanation Given: The force is f = m a , F= G m M r 3 r Assume that each planet moves in an orbit given by the vector valued function. Formula Used: The force is equal to F = − G m M r 5 r Proof: Since it is assumed that each planet moves in an orbit given by the vector valued function. m a = G m M r 3 r a = − G M

11th Edition

ISBN: 9781337275378

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Vector Arithmetic

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Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.

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Chapter 12.5, Problem 88E

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To Prove: The radius and acceleration are parallel for the vector valued function and then show that r(t)×r′(t)=L is a constant vector.

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Chapter 12.5, Problem 87E

Chapter 12.5, Problem 89E

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Chapter 12 Solutions

Multivariable Calculus

Ch. 12.1 - CONCEPTS CHECK Vector-valued FunctionDescribe how...Ch. 12.1 - Continuity of a Vector-valued FunctionDescribe...Ch. 12.1 - Prob. 3E Ch. 12.1 - Prob. 4E Ch. 12.1 - Prob. 5E Ch. 12.1 - Prob. 6E Ch. 12.1 - Prob. 7E Ch. 12.1 - Prob. 8E Ch. 12.1 - Prob. 9E Ch. 12.1 - Prob. 10E

Ch. 12.1 - Prob. 11E Ch. 12.1 - Prob. 12E Ch. 12.1 - Writing a Vector-Valued FunctionIn Exercises 1316,...Ch. 12.1 - Prob. 14E Ch. 12.1 - Writing a Vector-Valued FunctionIn Exercises 1316,...Ch. 12.1 - Prob. 16E Ch. 12.1 - Prob. 17E Ch. 12.1 - Prob. 18E Ch. 12.1 - Prob. 19E Ch. 12.1 - Prob. 20E Ch. 12.1 - Prob. 21E Ch. 12.1 - Prob. 22E Ch. 12.1 - Prob. 23E Ch. 12.1 - Prob. 24E Ch. 12.1 - Prob. 25E Ch. 12.1 - Prob. 26E Ch. 12.1 - Prob. 27E Ch. 12.1 - Prob. 28E Ch. 12.1 - Prob. 29E Ch. 12.1 - Prob. 30E Ch. 12.1 - Prob. 31E Ch. 12.1 - Prob. 32E Ch. 12.1 - Sketching a Space Curve In Exercises 31-38, sketch...Ch. 12.1 - Prob. 34E Ch. 12.1 - Prob. 35E Ch. 12.1 - Prob. 36E Ch. 12.1 - Prob. 37E Ch. 12.1 - Prob. 38E Ch. 12.1 - Prob. 39E Ch. 12.1 - Prob. 40E Ch. 12.1 - Prob. 41E Ch. 12.1 - Prob. 42E Ch. 12.1 - Prob. 43E Ch. 12.1 - Prob. 44E Ch. 12.1 - Prob. 45E Ch. 12.1 - Prob. 46E Ch. 12.1 - Representing a Graph by a Vector-Valued Function...Ch. 12.1 - Prob. 48E Ch. 12.1 - Representing a Graph by a Vector-Valued Function...Ch. 12.1 - Prob. 50E Ch. 12.1 - Prob. 51E Ch. 12.1 - Prob. 52E Ch. 12.1 - Representing a Graph by a Vector-Valued Function...Ch. 12.1 - Prob. 54E Ch. 12.1 - Prob. 55E Ch. 12.1 - Prob. 56E Ch. 12.1 - Prob. 57E Ch. 12.1 - Prob. 58E Ch. 12.1 - Prob. 59E Ch. 12.1 - Prob. 60E Ch. 12.1 - Prob. 61E Ch. 12.1 - Representing a Graph by Vector-Valued Function In...Ch. 12.1 - Prob. 63E Ch. 12.1 - Prob. 64E Ch. 12.1 - Finding a Limit In Exercises 65-70, find the limit...Ch. 12.1 - Prob. 66E Ch. 12.1 - Finding a Limit In Exercises 65-70, find the limit...Ch. 12.1 - Prob. 68E Ch. 12.1 - Finding a Limit In Exercises 65-70, find the limit...Ch. 12.1 - Prob. 70E Ch. 12.1 - Continuity of a Vector-Valued Function In...Ch. 12.1 - Prob. 72E Ch. 12.1 - Continuity of a Vector-Valued Function In...Ch. 12.1 - Prob. 74E Ch. 12.1 - Continuity of a Vector-Valued Function In...Ch. 12.1 - Prob. 76E Ch. 12.1 - Prob. 77E Ch. 12.1 - Prob. 78E Ch. 12.1 - Prob. 79E Ch. 12.1 - Prob. 80E Ch. 12.1 - Prob. 81E Ch. 12.1 - Prob. 82E Ch. 12.1 - Prob. 83E Ch. 12.1 - Prob. 84E Ch. 12.1 - Prob. 85E Ch. 12.1 - Prob. 86E Ch. 12.1 - Prob. 87E Ch. 12.1 - Prob. 88E Ch. 12.1 - Prob. 89E Ch. 12.1 - Prob. 90E Ch. 12.2 - CONCEPT CHECK Derivative Describe the relationship...Ch. 12.2 - Prob. 2E Ch. 12.2 - Prob. 3E Ch. 12.2 - Prob. 4E Ch. 12.2 - Differentiation of Vector-Valued FunctionsIn...Ch. 12.2 - Prob. 6E Ch. 12.2 - Differentiation of Vector-Valued FunctionsIn...Ch. 12.2 - Prob. 8E Ch. 12.2 - Prob. 9E Ch. 12.2 - Prob. 10E Ch. 12.2 - Prob. 11E Ch. 12.2 - Prob. 12E Ch. 12.2 - Prob. 13E Ch. 12.2 - Prob. 14E Ch. 12.2 - Prob. 15E Ch. 12.2 - Prob. 16E Ch. 12.2 - Prob. 17E Ch. 12.2 - Prob. 18E Ch. 12.2 - Prob. 19E Ch. 12.2 - Prob. 20E Ch. 12.2 - Higher-Order DifferentiationIn Exercises 1922,...Ch. 12.2 - Prob. 22E Ch. 12.2 - Higher-Order DifferentiationIn Exercises 2326,...Ch. 12.2 - Prob. 24E Ch. 12.2 - Higher-Order DifferentiationIn Exercises 2326,...Ch. 12.2 - Higher-Order DifferentiationIn Exercises 2326,...Ch. 12.2 - Prob. 27E Ch. 12.2 - Prob. 28E Ch. 12.2 - Finding Intervals on Which a Curve Is Smooth In...Ch. 12.2 - Prob. 30E Ch. 12.2 - Finding Intervals on Which a Curve Is Smooth In...Ch. 12.2 - Prob. 32E Ch. 12.2 - Prob. 33E Ch. 12.2 - Prob. 34E Ch. 12.2 - Prob. 35E Ch. 12.2 - Prob. 36E Ch. 12.2 - Using Two MethodsIn Exercises 37 and 38, find (a)...Ch. 12.2 - Prob. 38E Ch. 12.2 - Finding an Indefinite Integral In Exercises 39-46,...Ch. 12.2 - Prob. 40E Ch. 12.2 - Finding an Indefinite Integral In Exercises 39-46,...Ch. 12.2 - Prob. 42E Ch. 12.2 - Prob. 43E Ch. 12.2 - Prob. 44E Ch. 12.2 - Prob. 45E Ch. 12.2 - Prob. 46E Ch. 12.2 - Prob. 47E Ch. 12.2 - Prob. 48E Ch. 12.2 - Prob. 49E Ch. 12.2 - Prob. 50E Ch. 12.2 - Prob. 51E Ch. 12.2 - Evaluating a Definite Integral In Exercises 47-52,...Ch. 12.2 - Prob. 53E Ch. 12.2 - Prob. 54E Ch. 12.2 - Prob. 55E Ch. 12.2 - Prob. 56E Ch. 12.2 - Prob. 57E Ch. 12.2 - Finding an Antiderivative In Exercises 53-58, find...Ch. 12.2 - Prob. 59E Ch. 12.2 - Think About It Find two vector-valued functions...Ch. 12.2 - Prob. 61E Ch. 12.2 - Prob. 62E Ch. 12.2 - Prob. 63E Ch. 12.2 - Prob. 64E Ch. 12.2 - Prob. 65E Ch. 12.2 - Prob. 66E Ch. 12.2 - Prob. 67E Ch. 12.2 - Prob. 68E Ch. 12.2 - Prob. 69E Ch. 12.2 - Particle MotionA particle moves in the yz-plane...Ch. 12.2 - Prob. 71E Ch. 12.2 - Prob. 72E Ch. 12.2 - Prob. 73E Ch. 12.2 - True or False? In Exercises 73-76, determine...Ch. 12.2 - Prob. 75E Ch. 12.2 - Prob. 76E Ch. 12.3 - Prob. 1E Ch. 12.3 - Prob. 2E Ch. 12.3 - Prob. 3E Ch. 12.3 - Prob. 4E Ch. 12.3 - Finding Velocity and Acceleration Along a Plane...Ch. 12.3 - Prob. 6E Ch. 12.3 - Finding Velocity and Acceleration Along a Plane...Ch. 12.3 - Prob. 8E Ch. 12.3 - Finding Velocity and Acceleration Along a Plane...Ch. 12.3 - Prob. 10E Ch. 12.3 - Finding Velocity and Acceleration Vectors in Space...Ch. 12.3 - Prob. 12E Ch. 12.3 - Finding Velocity and Acceleration Vectors in Space...Ch. 12.3 - Prob. 14E Ch. 12.3 - Finding Velocity and Acceleration Vectors in Space...Ch. 12.3 - Prob. 16E Ch. 12.3 - Finding Velocity and Acceleration Vectors in Space...Ch. 12.3 - Prob. 18E Ch. 12.3 - Prob. 19E Ch. 12.3 - Prob. 20E Ch. 12.3 - Finding a Position Vector by Integration In...Ch. 12.3 - Prob. 22E Ch. 12.3 - Prob. 23E Ch. 12.3 - Prob. 24E Ch. 12.3 - Prob. 25E Ch. 12.3 - Prob. 26E Ch. 12.3 - Prob. 27E Ch. 12.3 - Prob. 28E Ch. 12.3 - Prob. 29E Ch. 12.3 - Prob. 30E Ch. 12.3 - Prob. 31E Ch. 12.3 - Prob. 32E Ch. 12.3 - Prob. 33E Ch. 12.3 - Prob. 34E Ch. 12.3 - Prob. 35E Ch. 12.3 - Prob. 36E Ch. 12.3 - Prob. 37E Ch. 12.3 - Prob. 38E Ch. 12.3 - Prob. 39E Ch. 12.3 - Prob. 40E Ch. 12.3 - Prob. 41E Ch. 12.3 - Prob. 42E Ch. 12.3 - Prob. 43E Ch. 12.3 - Prob. 44E Ch. 12.3 - Prob. 45E Ch. 12.3 - Prob. 46E Ch. 12.3 - Prob. 47E Ch. 12.3 - Prob. 48E Ch. 12.3 - Prob. 49E Ch. 12.3 - Prob. 50E Ch. 12.3 - Circular Motion In Exercises 51 and 52, use the...Ch. 12.3 - Prob. 52E Ch. 12.3 - Prob. 53E Ch. 12.3 - Prob. 54E Ch. 12.3 - Prob. 55E Ch. 12.3 - Particle Motion Consider a particle moving on an...Ch. 12.3 - Prob. 57E Ch. 12.3 - Prob. 58E Ch. 12.3 - Prob. 59E Ch. 12.3 - Prob. 60E Ch. 12.3 - Prob. 61E Ch. 12.3 - Prob. 62E Ch. 12.3 - Prob. 63E Ch. 12.4 - Prob. 1E Ch. 12.4 - Prob. 2E Ch. 12.4 - Finding the Unit Tangent Vector In Exercises 3-8,...Ch. 12.4 - Prob. 4E Ch. 12.4 - Finding the Unit Tangent Vector In Exercises 3-8,...Ch. 12.4 - Prob. 6E Ch. 12.4 - Finding the Unit Tangent Vector In Exercises 3-8,...Ch. 12.4 - Prob. 8E Ch. 12.4 - Prob. 9E Ch. 12.4 - Prob. 10E Ch. 12.4 - Prob. 11E Ch. 12.4 - Prob. 12E Ch. 12.4 - Prob. 13E Ch. 12.4 - Prob. 14E Ch. 12.4 - Finding the Principal Unit Normal Vector In...Ch. 12.4 - Prob. 16E Ch. 12.4 - Finding the Principal Unit Normal Vector In...Ch. 12.4 - Prob. 18E Ch. 12.4 - Prob. 19E Ch. 12.4 - Prob. 20E Ch. 12.4 - Prob. 21E Ch. 12.4 - Prob. 22E Ch. 12.4 - Prob. 23E Ch. 12.4 - Prob. 24E Ch. 12.4 - Prob. 25E Ch. 12.4 - Prob. 26E Ch. 12.4 - Prob. 27E Ch. 12.4 - Prob. 28E Ch. 12.4 - Prob. 29E Ch. 12.4 - Prob. 30E Ch. 12.4 - Prob. 31E Ch. 12.4 - Circular MotionIn Exercises 3134, consider an...Ch. 12.4 - Prob. 33E Ch. 12.4 - Prob. 34E Ch. 12.4 - Prob. 35E Ch. 12.4 - Prob. 36E Ch. 12.4 - Prob. 37E Ch. 12.4 - Prob. 38E Ch. 12.4 - Finding Tangential and Normal Components of...Ch. 12.4 - Prob. 40E Ch. 12.4 - Prob. 41E Ch. 12.4 - Prob. 42E Ch. 12.4 - Prob. 43E Ch. 12.4 - Prob. 44E Ch. 12.4 - Prob. 45E Ch. 12.4 - Prob. 46E Ch. 12.4 - Prob. 47E Ch. 12.4 - Prob. 48E Ch. 12.4 - Prob. 49E Ch. 12.4 - Prob. 50E Ch. 12.4 - Prob. 51E Ch. 12.4 - Prob. 52E Ch. 12.4 - Prob. 53E Ch. 12.4 - Prob. 54E Ch. 12.4 - Prob. 55E Ch. 12.4 - Prob. 56E Ch. 12.4 - Prob. 57E Ch. 12.4 - Prob. 58E Ch. 12.4 - Prob. 59E Ch. 12.4 - Prob. 60E Ch. 12.4 - Prob. 61E Ch. 12.4 - Prob. 62E Ch. 12.4 - Prob. 63E Ch. 12.4 - Prob. 64E Ch. 12.4 - Prob. 65E Ch. 12.4 - Prob. 66E Ch. 12.4 - Prob. 67E Ch. 12.4 - Prob. 68E Ch. 12.4 - Prob. 69E Ch. 12.4 - Prob. 70E Ch. 12.4 - Prob. 71E Ch. 12.4 - Prob. 72E Ch. 12.4 - Prob. 73E Ch. 12.4 - Prob. 74E Ch. 12.4 - Prob. 75E Ch. 12.4 - Prob. 76E Ch. 12.5 - Curvature Consider points P and Q on a curve What...Ch. 12.5 - Arc Length Parameter Let r(t) be a space curse....Ch. 12.5 - Prob. 3E Ch. 12.5 - Prob. 4E Ch. 12.5 - Prob. 5E Ch. 12.5 - Prob. 6E Ch. 12.5 - Prob. 7E Ch. 12.5 - Prob. 8E Ch. 12.5 - Projectile Motion The position of a baseball. is...Ch. 12.5 - Prob. 10E Ch. 12.5 - Prob. 11E Ch. 12.5 - Prob. 12E Ch. 12.5 - Prob. 13E Ch. 12.5 - Prob. 14E Ch. 12.5 - Prob. 15E Ch. 12.5 - Prob. 16E Ch. 12.5 - Investigation Consider the graph of the...Ch. 12.5 - Prob. 18E Ch. 12.5 - Prob. 19E Ch. 12.5 - Prob. 20E Ch. 12.5 - Prob. 21E Ch. 12.5 - Prob. 22E Ch. 12.5 - Prob. 23E Ch. 12.5 - Prob. 24E Ch. 12.5 - Prob. 25E Ch. 12.5 - Prob. 26E Ch. 12.5 - Finding CurvatureIn Exercises 2328, find the...Ch. 12.5 - Prob. 28E Ch. 12.5 - Prob. 29E Ch. 12.5 - Prob. 30E Ch. 12.5 - Prob. 31E Ch. 12.5 - Prob. 32E Ch. 12.5 - Prob. 33E Ch. 12.5 - Prob. 34E Ch. 12.5 - Finding Curvature In Exercises 29-36, find the...Ch. 12.5 - Prob. 36E Ch. 12.5 - Prob. 37E Ch. 12.5 - Prob. 38E Ch. 12.5 - Prob. 39E Ch. 12.5 - Prob. 40E Ch. 12.5 - Prob. 41E Ch. 12.5 - Prob. 42E Ch. 12.5 - Prob. 43E Ch. 12.5 - Prob. 44E Ch. 12.5 - Prob. 45E Ch. 12.5 - Prob. 46E Ch. 12.5 - Prob. 47E Ch. 12.5 - Prob. 48E Ch. 12.5 - Prob. 49E Ch. 12.5 - Prob. 50E Ch. 12.5 - Prob. 51E Ch. 12.5 - Prob. 52E Ch. 12.5 - Prob. 53E Ch. 12.5 - Prob. 54E Ch. 12.5 - Prob. 55E Ch. 12.5 - Prob. 56E Ch. 12.5 - Prob. 57E Ch. 12.5 - Prob. 58E Ch. 12.5 - Prob. 59E Ch. 12.5 - Prob. 60E Ch. 12.5 - Prob. 61E Ch. 12.5 - Prob. 62E Ch. 12.5 - Prob. 63E Ch. 12.5 - Prob. 64E Ch. 12.5 - Prob. 65E Ch. 12.5 - Speed The smaller the curvature of a bend in a...Ch. 12.5 - Prob. 67E Ch. 12.5 - Center of Curvature Use the result of Exercise 67...Ch. 12.5 - Prob. 69E Ch. 12.5 - Prob. 70E Ch. 12.5 - Prob. 71E Ch. 12.5 - Prob. 72E Ch. 12.5 - Prob. 73E Ch. 12.5 - Prob. 74E Ch. 12.5 - Prob. 75E Ch. 12.5 - Prob. 76E Ch. 12.5 - Curvature of a Cycloid Use the result of Exercise...Ch. 12.5 - Tangential and Normal Components of Acceleration...Ch. 12.5 - Prob. 79E Ch. 12.5 - Prob. 80E Ch. 12.5 - CurvatureVerify that the curvature at any point...Ch. 12.5 - Prob. 82E Ch. 12.5 - Prob. 83E Ch. 12.5 - Prob. 84E Ch. 12.5 - Prob. 85E Ch. 12.5 - Prob. 86E Ch. 12.5 - Prob. 87E Ch. 12.5 - Prob. 88E Ch. 12.5 - Prob. 89E Ch. 12.5 - Prob. 90E Ch. 12.5 - Prob. 91E Ch. 12.5 - Prob. 92E Ch. 12.5 - Prob. 93E Ch. 12.5 - Prob. 94E Ch. 12 - Domain and Continuity In Exercises 1-4, (a) find...Ch. 12 - Prob. 2RE Ch. 12 - Domain and Continuity In Exercises 1-4, (a) find...Ch. 12 - Domain and Continuity In Exercises 1-4, (a) find...Ch. 12 - Prob. 5RE Ch. 12 - Prob. 6RE Ch. 12 - Prob. 7RE Ch. 12 - Prob. 8RE Ch. 12 - Prob. 9RE Ch. 12 - Prob. 10RE Ch. 12 - Sketching a Curve In Exercises 9-12, sketch the...Ch. 12 - Sketching a Curve In Exercises 9-12, sketch the...Ch. 12 - Prob. 13RE Ch. 12 - Prob. 14RE Ch. 12 - Representing a Graph by a Vector-Valued Function...Ch. 12 - Representing a Graph by a Vector-Valued Function...Ch. 12 - Prob. 17RE Ch. 12 - Finding a Limit In Exercises 17 and 18, find the...Ch. 12 - Prob. 19RE Ch. 12 - Prob. 20RE Ch. 12 - Higher-Order Differentiation In Exercise 21 and...Ch. 12 - Prob. 22RE Ch. 12 - Prob. 23RE Ch. 12 - Finding Intervals on Which a Curve is SmoothIn...Ch. 12 - Prob. 25RE Ch. 12 - Prob. 26RE Ch. 12 - Prob. 27RE Ch. 12 - Prob. 28RE Ch. 12 - Prob. 29RE Ch. 12 - Prob. 30RE Ch. 12 - Prob. 31RE Ch. 12 - Prob. 32RE Ch. 12 - Prob. 33RE Ch. 12 - Prob. 34RE Ch. 12 - Prob. 35RE Ch. 12 - Prob. 36RE Ch. 12 - Prob. 37RE Ch. 12 - Prob. 38RE Ch. 12 - Prob. 39RE Ch. 12 - Prob. 40RE Ch. 12 - Prob. 41RE Ch. 12 - Projectile Motion In Exercises 41 and 42, use the...Ch. 12 - Finding the Unit Tangent Vector In Exercises 43...Ch. 12 - Prob. 44RE Ch. 12 - Prob. 45RE Ch. 12 - Prob. 46RE Ch. 12 - Prob. 47RE Ch. 12 - Prob. 48RE Ch. 12 - Prob. 49RE Ch. 12 - Prob. 50RE Ch. 12 - Prob. 51RE Ch. 12 - Prob. 52RE Ch. 12 - Prob. 53RE Ch. 12 - Finding Tangential and Normal Components of...Ch. 12 - Prob. 55RE Ch. 12 - Prob. 56RE Ch. 12 - Prob. 57RE Ch. 12 - Prob. 58RE Ch. 12 - Prob. 59RE Ch. 12 - Prob. 60RE Ch. 12 - Prob. 61RE Ch. 12 - Prob. 62RE Ch. 12 - Prob. 63RE Ch. 12 - Prob. 64RE Ch. 12 - Finding CurvatureIn Exercises 6366, find the...Ch. 12 - Finding CurvatureIn Exercises 6366, find the...Ch. 12 - Finding Curvature In Exercises 67 and 68, find the...Ch. 12 - Prob. 68RE Ch. 12 - Prob. 69RE Ch. 12 - Prob. 70RE Ch. 12 - Prob. 71RE Ch. 12 - Prob. 72RE Ch. 12 - Prob. 73RE Ch. 12 - Cornu Spiral The cornu spiral is given by...Ch. 12 - Prob. 2PS Ch. 12 - Prob. 3PS Ch. 12 - Prob. 4PS Ch. 12 - Cycloid Consider one arch of the cycloid...Ch. 12 - Prob. 6PS Ch. 12 - Prob. 7PS Ch. 12 - Prob. 8PS Ch. 12 - Binormal VectorIn Exercises 911, use the binormal...Ch. 12 - Prob. 10PS Ch. 12 - Prob. 11PS Ch. 12 - Prob. 12PS Ch. 12 - Prob. 13PS Ch. 12 - Ferris Wheel You want to toss an object to a...

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To Prove: The radius and acceleration are parallel for the vector valued function and then show that r ( t ) × r ′ ( t ) = L is a constant vector.

Solution Summary: The author explains that the acceleration a and radius r are parallel for the vector valued function.

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To Prove: The radius and acceleration are parallel for the vector valued function and then show that r(t)×r′(t)=L is a constant vector.

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Chapter 12 Solutions