Question

If angle between the line joining the points $(2,1,3)$ and $(1,-1,2)$ and the line having direction ratios $2,1,-1is\frac{2\pi }{m}$.Find $m$

Answer 1

The points are $(2,1,3)$ and $(1,-1,2)$

$\therefore$ DR of the line passing through this point : $(-1,-2,-1)$

DR of the another line : $(2,1,-1)$

$\therefore cos\theta =\frac{\left|\right(a_1,a_2+b_1{b}_2+c_2{c}_2\left)\right|}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$

$=\frac{-2-2+1}{\sqrt{1+4+1}\sqrt{1+4+1}}$

$=\frac{-3}{6}$

$=\frac{-1}{2}$

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

$\therefore \frac{2\pi }{3}=\frac{2\pi }{m}$

$m=3$