Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

111.(B) Let $\bar{a}=3\hat{i}-2\hat{j}+4\hat{k}$ and $\bar{b}=1\hat{i}+3\hat{j}-2\hat{k}$ $\begin{array}{rl}\bar{a}\times \bar{b}& =\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 3& -2& 4\\ 1& 3& -2\end{array}\right|\\ \bar{a}\times \bar{b}& =(4-12)\hat{i}-(-6-4)\hat{j}+(9+2)\hat{k}\\ & =-8\hat{i}+10\hat{j}+11\hat{k}\\ |\bar{a}\times \bar{b}|& =\sqrt{(-8{)}^2+(10{)}^2+(11{)}^2}=\sqrt{285}\end{array}$ Direction cosines are : $\frac{-8}{\sqrt{285}},\frac{10}{\sqrt{285}},\frac{11}{\sqrt{285}}$