Question

Given two vectors are $\hat{i}-\hat{j}$ and $\hat{i}+2\hat{j}$, then unit vector coplanar with the two vectors and perpendicular to first is

• A. $\frac{1}{\sqrt{2}}(\hat{i}+\hat{k})$
• B. $\frac{1}{\sqrt{5}}(2\hat{i}+\hat{j})$
• C. $\pm \frac{1}{\sqrt{2}}(\hat{i}+\hat{j})$
• D. None of these

Answer 1

Answer: (C)

$\pm \frac{1}{\sqrt{2}}(\hat{i}+\hat{j})$

Let $\overrightarrow{a}=\hat{i}-\hat{j}$ and $\overrightarrow{b}=\hat{i}+2\hat{j}$
The required vector is along the vector
$\overrightarrow{a}\times (\overrightarrow{a}\times \overrightarrow{b})=(\overrightarrow{a}.\overrightarrow{b})\overrightarrow{a}-(\overrightarrow{a}.\overrightarrow{a})\overrightarrow{b}$
$=-(\hat{i}-\hat{j})-2(\hat{i}+2\hat{j})$
$=-3\hat{i}-3\hat{j}$
Hence required vectors are given by
$\pm \frac{(-3\hat{i}-3\hat{j})}{\sqrt{9+9}}=\pm \frac{1}{\sqrt{2}}(\hat{i}+\hat{j})$