Q. 69 Let r(t) be a differentiable vec... [FREE SOLUTION]

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Chapter 11: Q. 69 (page 874)

Let $r (t)$ be a differentiable vector function such that $r (t) 路 r^{'} (t) = 0$ for every value of $t$ . Prove that $鈥� r (t) 鈥�$ is a constant.

Short Answer

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Ans:

$\frac{d}{d t} (鈥� r {鈥�}^{2}) = \frac{d}{d t} (r 路 r) = r \frac{d}{d t} (r) + r \frac{d}{d t} (r) = 0$

Step by step solution

Step 1. GIven information:

Consider a differentiable vector function $r (t)$ such that $r (t) 路 r^{'} (t) = 0$ For every value of $t$ .

Step 2. Proving:

Consider

$\frac{d}{d t} (鈥� r {鈥�}^{2}) = \frac{d}{d t} (r 路 r) s i c n e 鈥� r {鈥�}^{2} = r 路 r = r \frac{d}{d t} (r) + r \frac{d}{d t} (r) Derivativeforproductof 2 functions$

$\begin{array}{l} = r 路 r^{'} + r 路 r^{'} & \frac{d r}{d r} = r^{'} \\ = 2 r 路 r^{'} & add \\ = 2 (0) & since r 路 r^{'} = 0 \\ = 0 & Multiply \end{array} \frac{d}{d t} (鈥� r {鈥�}^{2}) = 0 . That means 鈥� r 鈥� is a constant, since the derivative of a constant is zero.$

Thus $鈥� r (t) 鈥�$ is a constant if $r (t) 路 r^{'} (t) = 0$

Note: The converse is always true. That is, if $r (t)$ is a differentiable vector function such that $鈥� r (t) 鈥�$ is a constant then $r (t) 路 r^{'} (t) = 0$ .

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91影视

Let $r (t)$ be a differentiable vector function such that $r (t) 路 r^{'} (t) = 0$ for every value of $t$ . Prove that $鈥� r (t) 鈥�$ is a constant.

Short Answer

Step by step solution

Step 1. GIven information:

Step 2. Proving:

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91影视

Let r(t) be a differentiable vector function such that r(t)路r'(t)=0 for every value of t. Prove that 鈥�r(t)鈥� is a constant.

Short Answer

Step by step solution

Step 1. GIven information:

Step 2. Proving:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Calculus

Decision Maths

Mechanics Maths

Probability and Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

Let $r (t)$ be a differentiable vector function such that $r (t) 路 r^{'} (t) = 0$ for every value of $t$ . Prove that $鈥� r (t) 鈥�$ is a constant.