Question

Find the angle between the lines whose direction ratios are $a,b,c$ and $b-c,c-a,a-b$

Answer 1

The angle $\theta$ between the lines with direction cosines $a,b,c$ and $b-c,c-a,a-b$ is given by,
$\cos \theta =\left|\frac{a(b-c)+b(c-a)+c(a-b)}{\sqrt{a^2+b^2+c^2}\sqrt{{(b-c)}^2+{(c-a)}^2+{(a-b)}^2}}\right|$
$\Rightarrow \cos \theta =0$
$\Rightarrow \theta ={\cos }^{-1}0$

$\Rightarrow \theta ={90}^o$
Thus, the angle between the lines is ${90}^o$.