Question

A unit vector perpendicular to the plane containing the vector $\hat{i}+2\hat{j}+\hat{k}$ and $-2\hat{i}+\hat{j}+3\hat{k}$ is

• A. $\frac{-\hat{i}+\hat{j}-\hat{k}}{\sqrt{3}}$
• B. $\frac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}$
• C. $\frac{-\hat{i}-\hat{j}-\hat{k}}{\sqrt{3}}$
• D. $\frac{-\hat{i}+\hat{j}-\hat{k}}{\sqrt{3}}$

Answer 1

The correct option is A $\frac{-\hat{i}+\hat{j}-\hat{k}}{\sqrt{3}}$

A unit vector perpendicular to the plane containing the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$
$=\hat{n}=\frac{\overrightarrow{a}\times \overrightarrow{b}}{|\overrightarrow{a}\times \overrightarrow{b}|}$
$\overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 2& 1\\ -2& 1& 3\end{array}\right|=5\hat{i}-5\hat{j}+5\hat{k}$
$|\overrightarrow{a}\times \overrightarrow{b}|=5\sqrt{3}$
$\therefore \hat{n}=\frac{\hat{i}-\hat{j}+\hat{k}}{\sqrt{3}}$ or, $\hat{n}=\frac{-\hat{i}+\hat{j}-\hat{k}}{\sqrt{3}}$