Explanation Given: The provided position vector is r → ( t ) = x ( t ) i ^ + y ( t ) j ^ + z ( t ) k ^ . Formula used: The magnitude of the velocity vector gives speed: ‖ v → ( t ) ‖ = x ' ( t ) 2 + y ' ( t ) 2 + z ' ( t ) 2 Proof: Consider the position vector, r → ( t ) = x ( t ) i ^ + y ( t ) j ^ + z ( t ) k ^ Differentiate position vector with respect to t gives a velocity vector, v → ( t ) = r → ' ( t ) = x ' ( t ) i ^ + y ' ( t ) j ^ + z ' ( t ) k ^ Differentiate velocity vector with respect to t gives an acceleration vector, a → ( t ) = r → " ( t ) = x " ( t ) i ^ + y " ( t ) j ^ + z " ( t ) k ^ It is provided that speed is constant

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Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 12.3, Problem 59E

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To prove: When an object is traveling at a constant speed, its velocity and acceleration vectors are orthogonal for the provided position vector r→(t)=x(t)i^+y(t)j^+z(t)k^.

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Chapter 12.3, Problem 58E

Chapter 12.3, Problem 60E

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Chapter 12 Solutions

Multivariable Calculus

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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? 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To prove: When an object is traveling at a constant speed, its velocity and acceleration vectors are orthogonal for the provided position vector r → ( t ) = x ( t ) i ^ + y ( t ) j ^ + z ( t ) k ^ .

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To prove: When an object is traveling at a constant speed, its velocity and acceleration vectors are orthogonal for the provided position vector r→(t)=x(t)i^+y(t)j^+z(t)k^.

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Chapter 12 Solutions