Question

Which one of the following is the unit vector perpendicular to both $\overrightarrow{a}=-\hat{i}+\hat{j}+\hat{k}$ and $\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}$?

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

Answer 1

Answer: (A)

$\frac{\hat{i}+\hat{j}}{\sqrt{2}}$

Let $A=-\hat{i}+\hat{j}+\hat{k}$

$B=\hat{i}-\hat{j}+\hat{k}$

Consider C = Unit vector

Now $C=A\times B$

$C=(-\hat{i}+\hat{j}+\hat{k})\times (\hat{i}-\hat{j}+\hat{k})$

$C=-2i+2j$

Hence unit vector$C=\frac{C}{\left|C\right|}$

$C=\frac{(-2i+2j)}{\sqrt{4}}$

Finally the unit vector $C=\frac{(i+j)}{\sqrt{2}}$.