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Q.
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
NTA AbhyasNTA Abhyas 2020
Solution:
Given, $l+m+n=0$ ...(i)
and $lm=0$
i.e. either $m=0$ or $l=0$
If $l=0$ , then putting in equation (i), we get $m=-n$
$\therefore $ Direction ratios are $0,-n,n$ i.e. $0,-1,1$
If $m=0$ , then putting in equation (i), we get $l=-n$
$\therefore $ Direction ratios are $ \, -n,0,n$ i.e. $ \, -1,0,1$
$\therefore \, \, cos \theta =\frac{0 \times \left(- 1\right) + \left(- 1\right) \times 0 + 1 \times 1}{\sqrt{\left(0\right)^{2} + \left(- 1\right)^{2} + \left(1\right)^{2}} \sqrt{\left(- 1\right)^{2} + \left(0\right)^{2} + \left(1\right)^{2}}}=\frac{1}{2}$
$\Rightarrow \, \, \theta =\frac{\pi }{3}$