The angle between the lines whose direction cosines satisfy the e…

MCQ

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

A
$\frac{\pi}{6}$
B
$\frac{\pi}{2}$
✓
$\frac{\pi}{3}$
D
$\frac{\pi}{4}$

✓

Answer

Correct option: C.

$\frac{\pi}{3}$

(C) Putting $l=- m - n$ in $l^2= m ^2+ n ^2$, we get $(- m - n )^2= m ^2+ n ^2$
$\Rightarrow m n=0 \Rightarrow m=0$ or $n=0$
If $m =0$, then $l=- n$
$\therefore \quad \frac{l}{-1}=\frac{ m }{0}=\frac{ n }{1}$
If $n =0$, then $l=- m$
$\therefore \quad \frac{l}{-1}=\frac{ m }{1}=\frac{ n }{0}$
$\therefore \quad a _1, b_1, c _1=-1,0,1$ and $a _2, b_2, c _2=-1,1,0$
$\therefore \quad$ The angle between the lines is given by
$\cos \theta=\frac{1+0+0}{\sqrt{1+0+1} \sqrt{1+1+0}}=\frac{1}{2}$
$\therefore \quad \theta=\frac{\pi}{3}$

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