Question

Find the angle between the lines where direction ratios are (a, b, c) and (b - c, c - a, a - b).

Answer 1

The direction ratios of the two lines are (a, b, c) and (b - c, c - a, a -b).
Let $\theta$ be the acute angle between the two lines, then
$cos\theta =\left|\frac{a(b-c)+b(c-a)+c(a-b)}{\sqrt{a^2+b^2+c^2}\sqrt{(b-c{)}^2+(c-a{)}^2+(a-b{)}^2}}\right|\because cos\theta =\left|\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}\right|=\left|\frac{ab-ac+bc-ab+ca-bc}{\sqrt{a^2+b^2+c^2}\sqrt{(b-c{)}^2+(c-a{)}^2+(a-b{)}^2}}\right|=0\Rightarrow cos\theta =0\Rightarrow \theta =\frac{\pi }{2}={90}^{\circ }$