A vector $\vec{r}$ of magnitude 4 makes an angle $\frac{\pi}{3}$ with the $x$-axis, an angle $\frac{2\pi}{3}$ with the $y$-axis and an obtuse angle with the $z$-axis. What is $\vec{r}$?

Explanation

1. 01

   $l = \cos\frac{\pi}{3} = \frac{1}{2}$ and $m = \cos\frac{2\pi}{3} = -\frac{1}{2}$, so $n^{2} = 1 - \frac{1}{4} - \frac{1}{4} = \frac{1}{2}$.

2. 02

   The angle with the $z$-axis is obtuse, so $n = -\frac{1}{\sqrt{2}}$.

3. 03

   $\vec{r} = 4\left(\frac{1}{2}\hat{i} - \frac{1}{2}\hat{j} - \frac{1}{\sqrt{2}}\hat{k}\right) = 2\hat{i} - 2\hat{j} - 2\sqrt{2}\hat{k}$.