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: 36


The rectangular coordinates are $(-\frac{5}{2},\frac{5}{2},\frac{5\sqrt 2}{2})$.

Work Step by Step

Since $\rho=5, \, \theta=\frac{3\pi}{4}, \, \phi=\frac{\pi}{4}$ and $$ x=\rho\cos \theta\sin \phi,\quad y=\rho\sin\theta\sin\phi, \quad z=\rho\cos \phi,$$ then we have $$ x=5\cos \frac{3\pi}{4}\sin \frac{\pi}{4}=-\frac{5}{2}, $$ $$ y=5\sin\frac{3\pi}{4}\sin \frac{\pi}{4}=\frac{5}{2}, $$ $$ z=5\cos \frac{\pi}{4}=\frac{5\sqrt 2}{2}.$$ Hence, the rectangular coordinates are $(-\frac{5}{2},\frac{5}{2},\frac{5\sqrt 2}{2})$.
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