The component of a vector $\mathbf{P}=3 \hat{i}+4 \hat{j}$ alog the direction $(\hat{i}+2 \hat{j})$ is

• A

  $\frac{8}{\sqrt{5}}$
• Correct answer
  B

  $\frac{11}{\sqrt{5}}$
• C

  $\frac{11}{2}$
• D

  $\sqrt{10}$

Solution

Given, $\mathbf{P}=3 \hat{i}+4 \hat{j}$ Let $Q=\hat{i}+2 \hat{j}$
The component of $\mathbf{P}$ along the $\mathbf{Q}$ is given as $|\mathbf{P}| \cos \theta$ Hence, we know that, $\mathbf{P} \cdot \mathbf{Q}=|\mathbf{P}||\mathbf{Q}| \cos \theta$ $\Rightarrow|P| \cos \theta=\frac{\mathbf{P} \cdot \mathbf{Q}}{|\mathbf{Q}|}$ $=\frac{(3 \hat{i}+4 \hat{j}) \cdot(\hat{i}+2 \hat{j})}{\sqrt{1^2+2^2}}=\frac{3+8}{\sqrt{5}}=\frac{11}{\sqrt{5}}$

Hence, the correct answer is option: 'B'.