The direction cosines $l, m, n$ of two lines satisfy $2l + 2m - 3n = 0$ and $mn + nl - 3lm = 0$. What is the cosine of the angle between the lines?

Explanation

1. 01

   From the first, $n = \frac{2(l + m)}{3}$; then $mn + nl = n(l + m) = 3lm$ gives $2(l + m)^{2} = 9lm$, that is $(2l - m)(l - 2m) = 0$.

2. 02

   $m = 2l$ gives direction ratios $1, 2, 2$; $l = 2m$ gives $2, 1, 2$.

3. 03

   $\cos \theta = \frac{2 + 2 + 4}{3 \times 3} = \frac{8}{9}$.