Question

The acute angle between two lines whose direction cosines are given by the relation between $l+m+n=0$ and $l^2+m^2-n^2=0$ is

• A. $\frac{\pi }{2}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{4}$
• D. $\frac{\pi }{6}$

Answer 1

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

$l+m+n=0$ ……..(i)
$l^2+m^2-n^2=0$ ……… (ii)
From Eqs. (i) and (ii),
$l^2+m^2-(-l-m{)}^2=0$
$lm=0$
$\therefore l=0orm=0$

Case I : If, l = 0
Then, m + n = 0
$\therefore \frac{l}{0}=\frac{m}{-1}=\frac{n}{1}=\frac{1}{\sqrt{2}}$

Case II : If, m = 0
Then, l+n=0
$\therefore \frac{l}{-1}=\frac{m}{0}=\frac{n}{1}=\frac{1}{\sqrt{2}}$
If angle between lines is \theta
$\therefore cos\theta =0+0+\frac{1}{2}$
$\therefore \theta =\frac{\pi }{3}$