The direction cosines $l$, $m$ and $n$ of two lines are connected by the relations $l + m + n = 0$ and $lm = 0$, then the angle between the lines is
Step-by-Step Solution
Key Concept: Use the constraint equations on direction cosines to find possible direction ratios, then apply the angle formula between two lines.
Given $l + m + n = 0$ and $lm = 0$, either $l = 0$ or $m = 0$. If $l = 0$, then putting in equation (i), we get $n = -n$, so direction ratios are $0, -n, n$ which gives $0, -1, 1$. If $m = 0$, then putting in equation (i), we get $l = -n$, so direction ratios are $-n, 0, n$ which gives $-1, 0, 1$. Computing $\cos\theta = \frac{|0(-1) + (-1)(0) + 1(1)|}{\sqrt{0^2+1^2+1^2}\sqrt{1^2+0^2+1^2}} = \frac{1}{2}$, giving $\theta = \frac{\pi}{3}$.
Correct Answer: 2