Question

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

Answer 1

$\begin{array}{c}\left(1,-1,2\right)\equiv \hat{i}-\hat{j}+2\hat{k}\equiv \overrightarrow{a}\\ \left(2,1,-1\right)\equiv 2\hat{i}+\hat{j}-\hat{k}\equiv \overrightarrow{b}\\ \overrightarrow{c}=\overrightarrow{a}\times \overrightarrow{b}\\ =\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& -1& 2\\ 2& 1& -1\end{array}\right|\\ =\hat{i}-\hat{j}\left(-5\right)+\hat{k}\left(3\right)\\ =-\hat{i}+5\hat{j}+3\hat{k}\\ Thus,direction\cos ineoflineperpendicularto\overrightarrow{a}and\overrightarrow{b}is\\ \left(\frac{-1}{\sqrt{35}},\frac{5}{\sqrt{35}},\frac{3}{\sqrt{35}}\right)\end{array}$