Question

A unit vector perpendicular to the plane containing the vectors $\hat{i} + 2 \hat{j} + \hat{k}$ 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

Answer: (A)

$\frac{- \hat{i} + \hat{j} - \hat{k}}{\sqrt{3}}$

Answer 2

$\hat{n}=\left(\frac{\vec{a}\times\vec{b}}{\left|\vec{a}\times\vec{b}\right|}\right)$
$\vec{a}\times\vec{b}=\left|\begin{matrix}\hat{i}&\hat{j}&\hat{k}\\\ 1&2&1\\\ -2&1&3\end{matrix}\right|=5\hat{i}-5\hat{j}+5\hat{k}$
$\left|\vec{a}\times\vec{b}\right|=5\sqrt{3}$
$\therefore\, \hat{n}=\frac{\hat{i}-\hat{j}+\hat{k}}{\sqrt{3}}$ or $\hat{n}=\frac{-i+j-\hat{k}}{\sqrt{3}}$