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

The angle Q between the lines with direction cosines, a, b, c and b − c, c − a,a − b, is given by,

$\cos Q=\left|\frac{a(b-c)+b(c-a)+c(a-b)}{\sqrt{a^{2}+b^{2}+c^{2}}+\sqrt{(b-c)^{2}+(c-a)^{2}+(a-b)^{2}}}\right|$

$\Rightarrow \cos Q=0$

$\Rightarrow Q=\cos ^{-1} 0$

$\Rightarrow Q=90^{\circ}$

Thus, the angle between the lines is 90°.