Question

Find out the unit vector perpendicular to both vectors $\hat{i}-\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+\hat{k}$ ?

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

Answer 1

The correct option is B $\frac{-\hat{i}+\hat{k}}{\sqrt{2}}$

$\overrightarrow{a}=\hat{i}-\hat{j}+\hat{k}\overrightarrow{b}=\hat{i}+\hat{j}+\hat{k}$

Let $\hat{n}$ be a unit vector perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{b}$.

Thus, $\hat{n}=\frac{\overrightarrow{a}\times \overrightarrow{b}}{|\overrightarrow{a}\times \overrightarrow{b}|}$

$|\overrightarrow{a}\times \overrightarrow{b}|=(\hat{i}-\hat{j}+\hat{k})\times (\hat{i}+\hat{j}+\hat{k})=\hat{k}-\hat{j}+\hat{k}-\hat{i}+\hat{j}-\hat{i}=2(-\hat{i}+\hat{k})\Rightarrow \hat{n}=\frac{2(-\hat{i}+\hat{k})}{2\sqrt{2}}=\frac{(-\hat{i}+\hat{k})}{\sqrt{2}}$