Question

The direction cosines $l,m,n$ of two lines are connected by the relations $l+m+n=0$, $lm=0$, then the angle between them is

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{2}$
• D. $0$

Answer 1

The correct option is A $\frac{\pi }{3}$

Given, $l+m+n=\mathrm{0.....}\left(i\right)$

And $lm=0$

Either, $m=0$ or $l=0$

If $l=0$, then put in eq (i) we get $m=-n$

$\therefore$ Direction ratios are $0.-n,n$ i.e., $0,-1,1$

If $m=0$, then put in eq.(i) we get $l=-n$

$\therefore$ Direction ratios are $-n,0,n$ i.e.,$-1,0,1$

$\cos \theta =\frac{0(-1)+(-1)\times 0+1\times 1}{\sqrt{{\left(0\right)}^2+{(-1)}^2+{\left(1\right)}^2}\sqrt{{(-1)}^2+{\left(0\right)}^2+{\left(1\right)}^2}}=\frac{1}{2}$

$\Rightarrow \theta =\frac{\pi }{3}$