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)

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