Question

A unit vector perpendicular to $ \hat{i}-\hat{j}+\hat{k} $ and $ \hat{i}+\hat{j}-\hat{k} $ is

• A. $ \frac{\hat{k}+\hat{i}}{\sqrt{2}} $
• B. $ \frac{\hat{j}+\hat{k}}{\sqrt{2}} $
• C. $ \frac{\hat{i}-\hat{k}}{\sqrt{3}} $
• D. $ \frac{\hat{j}-\hat{k}}{\sqrt{2}} $

Answer 1

Answer: (B)

Let $ \vec{a}=\hat{i}-\hat{j}+\hat{k} $ and $ \vec{b}=\hat{i}+\hat{j}-\hat{k} $
Now, $ \vec{a}\times \vec{b}=\left| \begin{matrix} {\hat{i}} & {\hat{j}} & {\hat{k}} \\ 1 & -1 & 1 \\ 1 & 1 & -1 \\ \end{matrix} \right| $
$ =\hat{i}(1-1)-\hat{j}(-1-1)+\hat{k}(1+1) $
$ =2\hat{j}+2\hat{k} $
$ \therefore $ Required unit vector
$ =\pm \frac{2\hat{j}+2\hat{k}}{\sqrt{4+4}} $
$ =\pm \frac{\hat{j}+\hat{k}}{\sqrt{2}} $