Question

Find the direction cosines of a line which is perpendicular to the lines whose direction ratios are $(1,-1,2)$ and $(2,1,-1).$

Answer 1

let

$a=i-j+2k$ and $b=2i+j-k$

$a\times b=\left|\begin{array}{ccc}i& j& k\\ 1& -1& 2\\ 2& 1& -1\end{array}\right|$

$a\times b=-i+5j+3k$

$|a\times b|=\sqrt{(-1{)}^2+5^2+3^2}=\sqrt{35}$

Direction cosines are :$(\pm \frac{1}{\sqrt{35}},\pm \frac{5}{\sqrt{35}},\pm \frac{3}{\sqrt{35}})$