The angle between the lines whose direction cosines are connected… - Vidyadip

MCQ

The angle between the lines whose direction cosines are connected by the relations $l + m + n = 0$ and $2lm + 2nl - mn = 0$, is

A
$\frac{\pi }{3}$
✓
$\frac{{2\pi }}{3}$
C
$\pi $
D
None of these

✓

Answer

Correct option: B.

$\frac{{2\pi }}{3}$

b
(b) Eliminating $n$, we have $(2l + m)\,(l - m) = 0$

When $2l + m = 0,$ then $\frac{l}{1} = \frac{m}{{ - 2}} = \frac{n}{1}$

When $l - m = 0,$ then $\frac{l}{1} = \frac{m}{1} = \frac{n}{{ - 2}}$

$\therefore $ Direction ratios are $1, -2, 1$ and $1, 1, -2.$

$\cos \theta = \frac{{\sum {a_1}{a_2}}}{{\sqrt {(\sum a_1^2)\,} .\sqrt {(\sum a_2^2)\,} }} = - \frac{1}{2}\,$

$ \Rightarrow \,\,\theta = {120^o} = \frac{{2\pi }}{3}.$

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