Question

The unit vector perpendicular to the plane passing through points $P\left(\hat{i}-\hat{j}+2\hat{k}\right),Q\left(2\hat{i}-\hat{k}\right)\mathrm{and}R\left(2\hat{j}+\hat{k}\right)$ is
(a) $2\hat{i}+\hat{j}+\hat{k}$

(b) $\sqrt{6}\left(2\hat{i}+\hat{j}+\hat{k}\right)$

(c) $\frac{1}{\sqrt{6}}\left(2\hat{i}+\hat{j}+\hat{k}\right)$

(d) $\frac{1}{6}\left(2\hat{i}+\hat{j}+\hat{k}\right)$

Answer 1

(c) $\frac{1}{\sqrt{6}}\left(2\hat{i}+\hat{j}+\hat{k}\right)$


$$\begin{gathered} \mathrm{The}\mathrm{vector}\overrightarrow{PQ}\times \overrightarrow{PR}\mathrm{is}\mathrm{perpendicular}\mathrm{to}\mathrm{the}\mathrm{vectors}\overrightarrow{PQ}\mathrm{and}\overrightarrow{PR}. \\ \therefore \text{Required unit vector=}=\frac{\overrightarrow{PQ}\times \overrightarrow{PR}}{\left|\overrightarrow{PQ}\times \overrightarrow{PR}\right|} \\ \mathrm{Now}, \\ \overrightarrow{PQ}=\mathrm{P}.\mathrm{V}.\mathrm{of}Q-\mathrm{P}.\mathrm{V}.\mathrm{of}P \\ =\hat{i}+\hat{j}-3\hat{k} \\ \overrightarrow{PR}=\mathrm{P}.\mathrm{V}.\mathrm{of}R-\mathrm{P}.\mathrm{V}.\mathrm{of}P \\ =-\hat{i}+3\hat{j}-\hat{k} \\ \therefore \overrightarrow{PQ}\times \overrightarrow{PR}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 1& -3\\ -1& 3& -1\end{array}\right| \\ =8\hat{i}+4\hat{j}+4\hat{k} \\ =4\left(2\hat{i}+\hat{j}+\hat{k}\right) \\ \Rightarrow \left|\overrightarrow{PQ}\times \overrightarrow{PR}\right|=\sqrt{64+16+16} \\ =\sqrt{96} \\ =4\sqrt{6} \\ \text{Required unit vector}=\frac{\overrightarrow{PQ}\times \overrightarrow{PR}}{\left|\overrightarrow{PQ}\times \overrightarrow{PR}\right|} \\ =\frac{4\left(2\hat{i}+\hat{j}+\hat{k}\right)}{4\sqrt{6}} \\ =\frac{1}{\sqrt{6}}\left(2\hat{i}+\hat{j}+\hat{k}\right) \end{gathered}$$