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

The given vectors are $\overrightarrow{a}=\hat{i}+2\hat{j}-3\hat{k}$ and $\overrightarrow{b}=3\hat{i}-\hat{j}+2\hat{k}$.

Unit vectors perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{b}$ = $\pm \frac{\overrightarrow{a}\times \overrightarrow{b}}{\left|\overrightarrow{a}\times \overrightarrow{b}\right|}$

Now,
$$\begin{gathered} \overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 2& -3\\ 3& -1& 2\end{array}\right|=\hat{i}-11\hat{j}-7\hat{k} \\ \therefore \left|\overrightarrow{a}\times \overrightarrow{b}\right|=\left|\hat{i}-11\hat{j}-7\hat{k}\right|=\sqrt{1^2+{\left(-11\right)}^2+{\left(-7\right)}^2}=\sqrt{1+121+49}=\sqrt{171} \end{gathered}$$

Unit vectors perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{b}$ = $\pm \frac{\hat{i}-11\hat{j}-7\hat{k}}{\sqrt{171}}$

∴ Required vectors = $\sqrt{171}\left(\pm \frac{\hat{i}-11\hat{j}-7\hat{k}}{\sqrt{171}}\right)=\pm \left(\hat{i}-11\hat{j}-7\hat{k}\right)$

Thus, the vectors of magnitude $\sqrt{171}$ which are perpendicular to both the given vectors are $\pm \left(\hat{i}-11\hat{j}-7\hat{k}\right)$.