A point $P$ is at a distance of 8 from the origin $O$. The line $OP$ makes angles $\frac{2\pi}{3}$ and $\frac{3\pi}{4}$ with the $x$- and $y$-axes and an acute angle with the $z$-axis. What are the coordinates of $P$?

Explanation

1. 01

   $l = \cos\frac{2\pi}{3} = -\frac{1}{2}$ and $m = \cos\frac{3\pi}{4} = -\frac{1}{\sqrt{2}}$, so $n^{2} = 1 - \frac{1}{4} - \frac{1}{2} = \frac{1}{4}$; the acute angle gives $n = \frac{1}{2}$.

2. 02

   $P = (8l, 8m, 8n) = (-4, -4\sqrt{2}, 4)$.