Question

The unit vector perpendicular to the vector $\hat{i}-\hat{j}$ and $\hat{i}+\hat{j}$ forming a right handed system, is

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

Answer 1

The correct option is A $\hat{k}$

Let $\overrightarrow{a}=\hat{i}-\hat{j}$
and $\overrightarrow{b}=\hat{i}+\hat{j}$
Let the unknown vector be $\overrightarrow{c}$
Given $\overrightarrow{c}\perp \overrightarrow{a}$ and $\overrightarrow{b}$ and $|\overrightarrow{c}|=1$
$\Rightarrow \overrightarrow{a}\times \overrightarrow{b}=(\hat{i}-\hat{j})\times (\hat{i}+\hat{j})=2\hat{k}$
$\overrightarrow{c}=\frac{\overrightarrow{a}\times \overrightarrow{b}}{|\overrightarrow{a}\times \overrightarrow{b}|}$
$\Rightarrow \hat{c}=\frac{2\hat{k}}{2}$
$\Rightarrow \hat{c}=\hat{k}$