Question

Find a vector of magnitude $\sqrt{171}$ which is perpendicular to both of the vectors $\overrightarrow{a}=\hat{i}+2\hat{j}-3\hat{k}$ and $\overrightarrow{b}=3\hat{i}-\hat{j}+2\hat{k}$.

Answer 1

We have $\overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 2& -3\\ 3& -1& 2\end{array}\right|=\hat{i}-11\hat{j}-7\hat{k}$ $\therefore$ Required vector = $\sqrt{171}(\pm \frac{\overrightarrow{a}\times \overrightarrow{b}}{|\overrightarrow{a}\times \overrightarrow{b}|})=\sqrt{171}(\pm \frac{\hat{i}-11\hat{j}-7\hat{k}}{\sqrt{1+121+49}})=\pm (\hat{i}-11\hat{j}-7\hat{k})$.