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, $\vec{a}×\vec{b}=â\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& −1& 1\\ 1& 1& −1\end{array}â$
$=\hat{i}(1−1)−\hat{j}(−1−1)+\hat{k}(1+1)$
$=2\hat{j}+2\hat{k}$
$∴$ Required unit vector
$=Â\pm \frac{2\hat{j}+2\hat{k}}{\sqrt{4+4}}$
$=Â\pm \frac{\hat{j}+\hat{k}}{\sqrt{2}}$