Question

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

• A. $\frac{(\hat{i}+\hat{j}-\hat{k})}{\sqrt{3}}$
• B. $\frac{(\hat{i}-\hat{j}+\hat{k})}{\sqrt{3}}$
• C. $\hat{i}+\hat{j}-\hat{k}$
• D. $\frac{4(\hat{i}+\hat{j}-\hat{k})}{\sqrt{3}}$

Answer 1

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

Vector perpendicular to both the vectors is given by
$\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& 1& 3\\ 2& -1& 1\end{array}\right|=4\hat{i}+4\hat{j}-4\hat{k}=4(\hat{i}+\hat{j}-\hat{k})$

Required unit vector is $\frac{4(\hat{i}+\hat{j}-\hat{k})}{4\sqrt{1^2+1^2+(-1{)}^2}}=\frac{(\hat{i}+\hat{j}-\hat{k})}{\sqrt{3}}$