Answer
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 $(-\frac{5}{2},\frac{5}{2},\frac{5\sqrt 2}{2})$.
