   Chapter 12.3, Problem 53E

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# Proof Prove that when an object is traveling at a constant speed, its velocity and acceleration vectors are orthogonal.

To determine

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^.

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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