Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 14 - Calculus of Vector-Valued Functions - 14.1 Vector-Valued Functions - Exercises - Page 712: 44


$$r(t)=\lt 3, 1+2\cos t, 5+3\sin t\gt.$$

Work Step by Step

We know that the equation of the given ellipse in Figure 15 B and centered at $(3,1,5)$ is $$ \left(\frac{y-1}{2}\right)^{2}+\left(\frac{z-5}{3}\right)^{2}=1 $$ To parametrize the ellipse, we put $\frac{y-1}{2}=\cos t$ and $\frac{z-5}{3}=\sin t$, so we have $y=1+2\cos t$ and $5=5+3\sin t.$ That is, $$r(t)=\lt 3, 1+2\cos t, 5+3\sin t\gt.$$
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