What is the angle between the lines $x = 2y = -z$ and $x = y = 2z$?

Explanation

1. 01

   Put each equal to $t$: the first line is $x = t$, $y = \frac{t}{2}$, $z = -t$, with direction ratios $1, \frac{1}{2}, -1$, or $2, 1, -2$.

2. 02

   Likewise the second has $1, 1, \frac{1}{2}$, or $2, 2, 1$.

3. 03

   $\cos \theta = \frac{\lvert 4 + 2 - 2 \rvert}{3 \times 3} = \frac{4}{9}$, so $\theta = \cos^{-1}\frac{4}{9}$.