Question

The unit vector perpendicular to $\hat{i}-\hat{j}$and coplanar with$\hat{i}+2\hat{j}$and$\hat{i}+3\hat{j}$ is?

• A. $\frac{2\hat{i}-5\hat{j}}{\sqrt{59}}$
• B. $2\hat{i}+5\hat{j}$
• C. $\frac{\hat{i}+\hat{j}}{\sqrt{2}}$
• D. $\hat{i}+\hat{j}$

Answer 1

The correct option is C

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

Explanation for correct option

Let the unit vector be $\hat{a}=x\hat{i}+y\hat{j}$

As per the given condition, a is perpendicular to vector $\hat{i}-\hat{j}$

Therefore,

$$\begin{gathered} \therefore \hat{a}.(\hat{i}-\hat{j})=0 \\ \Rightarrow (x\hat{i}+y\hat{j}).(\hat{i}-\hat{j})=0 \\ \Rightarrow x-y=0 \\ \Rightarrow x=y \end{gathered}$$

So, we can write,

$$\begin{gathered} \hat{a}=x\hat{i}+x\hat{j} \\ \Rightarrow \left|\hat{a}\right|=\sqrt{x^2+x^2} \\ =x\sqrt{2} \end{gathered}$$

Therefore, the unit vector $\hat{a}=\frac{x(\hat{i}+\hat{j})}{x\sqrt{2}}=\frac{\hat{i}+\hat{j}}{\sqrt{2}}$

Hence, the correct option is $\mathrm{Option}\left(\mathrm{C}\right)$