Calculus (3rd Edition)

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

Chapter 13 - Vector Geometry - 13.7 Cylindrical and Spherical Coordinates - Exercises - Page 700: 34


$(\frac{\sqrt 6}{2},\frac{\sqrt 6}{2},1)$.

Work Step by Step

Since $\rho=2, \, \theta=\frac{\pi}{4}, \, \phi=\frac{\pi}{3}$ and $$ x=\rho\cos \theta\sin \phi,\quad y=\rho\sin\theta\sin\phi, \quad z=\rho\cos \phi,$$ then we have $$ x=2\cos \frac{\pi}{4}\sin \frac{\pi}{3}=\frac{\sqrt 6}{2}, $$ $$ y=3\sin \frac{\pi}{4}\sin \frac{\pi}{3}=\frac{\sqrt 6}{2}, $$ $$ z=3\cos \frac{\pi}{3}=1.$$ Hence, the rectangular coordinates are $(\frac{\sqrt 6}{2},\frac{\sqrt 6}{2},1)$.
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