For which $k$ is the angle between the lines with direction ratios $2, -1, 2$ and $1, 1, k$ equal to $\frac{\pi}{4}$?

Explanation

1. 01

   $\cos\frac{\pi}{4} = \frac{\lvert 2 - 1 + 2k \rvert}{3\sqrt{2 + k^{2}}} = \frac{1}{\sqrt{2}}$, so $2(1 + 2k)^{2} = 9(2 + k^{2})$.

2. 02

   $8k^{2} + 8k + 2 = 9k^{2} + 18$ gives $k^{2} - 8k + 16 = 0$, that is $(k - 4)^{2} = 0$, so $k = 4$.