Question

A unit vector coplanar with $\overrightarrow{i}+\overrightarrow{j}+3\overrightarrow{k}\text{and}\overrightarrow{i}+3\overrightarrow{j}+\overrightarrow{k}$ and perpendicular to $\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k}$ is

• A. $\frac{1}{\sqrt{2}}(i+j)$
• B. $\frac{1}{\sqrt{3}}(\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k})$
• C. $\frac{1}{\sqrt{2}}(\overrightarrow{j}-\overrightarrow{k})$
• D. $\frac{1}{\sqrt{3}}(\overrightarrow{i}+\overrightarrow{j}-\overrightarrow{k})$

Answer 1

The correct option is C

$\frac{1}{\sqrt{2}}(\overrightarrow{j}-\overrightarrow{k})$

$\text{Let}\hat{a}=x\hat{i}+y\hat{j}+z\hat{k}\Rightarrow \left|\begin{array}{ccc}x& y& z\\ 1& 1& 3\\ 1& 3& 1\end{array}\right|=0\Rightarrow 4x-y-z=0\text{also}x+y+z=0\Rightarrow x=0\text{and}y=-z\therefore \hat{a}=y\hat{j}-y\hat{k}\text{is a unit vector}\Rightarrow y=\frac{1}{\sqrt{2}}\therefore \hat{a}=\frac{1}{\sqrt{2}}(\hat{j}-\hat{k})$