Question

Find the direction cosines of the line which is perpendicular to the lines which direction cosines proportional to 1, -2, -2 and 0, 2, 1.

Answer 1

Let $(a,b,c)$ be the direction cosine perpendicular to the lines $(1,-2,-2)$ and $(0,2,-1)$

$\Rightarrow a_1{a}_2+b_1{b}_2+c_1{c}_2=0$

$\Rightarrow a-2b-2c=0$ ...$\left(1\right)$

and $2b-c=0$ ..... $\left(2\right)$

$\frac{a}{-2+4}=\frac{-b}{1-0}=\frac{c}{2}$

$\Rightarrow \frac{a}{2}=\frac{b}{-1}=\frac{c}{2}$

Thus,$(2,-1,2)$ is the direction cosine of the line perpendicular to the above lines.