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Q.
The acute angle between two lines such that the direction cosines $l,m,n,$ of each of them satisfy the equations $l+m+n=0$ and $l^{2}+m^{2}-n^{2}=0$ is:
NTA AbhyasNTA Abhyas 2022
Solution:
Let $l_{1},m_{1},n_{1}$ and $l_{2},m_{2},n_{2}$ be the d.c of line $1$ and $2$ respectively and angle between lines, $\theta $ is
$cos\theta =l_{1}l_{2}+m_{1}m_{2}+n_{1}n_{2}$
Now, as given
$l^{2}+m^{2}=n^{2}$ and $l+m=-n$
$\Rightarrow \left(-n \right)^{2}-2lm=n^{2}\Rightarrow 2lm=0$ or $lm=0$
So the direction ratios of the two lines will be
$\left(0 , 1 , 1\right)\&\left(1 , 0 , 1\right)$
$\therefore cos\theta =\frac{1}{2}\Rightarrow \theta =60^\circ $ (acute angle)