Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 13: Vector-Valued Functions and Motion in Space - Section 13.3 Arc Length in Space - Exercises 13.3 - Page 760: 20


$$ \mathbf{T} =\cos t \mathbf{i}+\sin t \mathbf{j}$$

Work Step by Step

Since $$\mathbf{r}=(\cos t+t \sin t) \mathbf{i}+(\sin t+t \cos t) \mathbf{j} $$ Then \begin{align*} \mathbf{v}&=(-\sin t+t \cos t+\sin t) \mathbf{i}+(\cos t-(t(-\sin t)+\cos t)) \mathbf{j}\\ &=(t \cos t) \mathbf{i}+(t \sin t) \mathbf{j} \\ |\mathbf{v}|&=\sqrt{(t \cos t)^{2}+(t \sin t)^{2}}&\\ &=\sqrt{t^{2}}=|t|=t, \end{align*} Hence \begin{align*} \mathbf{T}&=\frac{\mathbf{v}}{|\mathbf{v}|}\\ &=\frac{t \cos t}{t} \mathbf{i}+\frac{t \sin t}{t} \mathbf{j}\\ &=\cos t \mathbf{i}+\sin t \mathbf{j}\end{align*}
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