Question

The angle between the lines whose direction cosines satisfy the equations $l+m+n=0$ and $l^2=m^2+n^2$ is?

Answer 1

Angle between two lines is:

$\begin{array}{c}\cos \theta =\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{{a_1}^2+{b_1}^2+{c_1}^2}\sqrt{{a_2}^2+{b_2}^2+{c_2}^2}}\\ l+m+n=\mathrm{0.......}\left(1\right)\\ l=-\left(m+n\right)\\ {\left(m+n\right)}^2=l^2........\left(2\right)\\ m^2+n^2+2mn=m^2+n^2\\ \left[l^2=m^2+n^2,given\right]\\ 2mn=0\\ Whenm=0\Rightarrow l=-n\\ Hence,\left(l,m,n\right)is\left(1,0,-1\right)\\ Whenn=0,thenl=-m\\ Hence,\left(l,m,n\right)is\left(1,0,-1\right)\\ \cos \theta =\frac{1+0+0}{\sqrt{2}\sqrt{2}}=\frac{1}{2}\Rightarrow \theta =\frac{\pi }{3}\end{array}$