Question

Write a unit vector perpendicular to both the vectors $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}$ and $\overrightarrow{b}=\hat{i}+\hat{j}.$

Answer 1

A unit vector in the direction $\overrightarrow{a}\times \overrightarrow{b}$ is perpendicular to both the vectors.

So,

$\overrightarrow{a}\times \overrightarrow{b}=(\hat{i}+\hat{j}+\hat{k})\times (\hat{i}+\hat{j})=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 1& 1\\ 1& 1& 0\end{array}\right|=-\hat{i}+\hat{j}$ and $|\overrightarrow{a}\times \overrightarrow{b}|=\sqrt{1^2+1^2}=\sqrt{2}.$

Thus, the required vector is $\pm \left(\frac{-\hat{i}+\hat{j}}{\sqrt{2}}\right).$