Question

The velocity of a particle is $\overrightarrow{v}=6\hat{i}+2\hat{j}-2\hat{k}.$ The component of the velocity parallel to vector $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}$ is :-

• A. $6\hat{i}+2\hat{j}+2\hat{k}$
• B. $2\hat{i}+2\hat{j}+2\hat{k}$
• C. $\hat{i}+\hat{j}+\hat{k}$
• D. $6\hat{i}+2\hat{j}-2\hat{k}$

Answer 1

The correct option is B $2\hat{i}+2\hat{j}+2\hat{k}$

$\overrightarrow{v}=6\hat{i}+2\hat{j}-2\hat{k}$

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

component $\overrightarrow{a}$ is $(\overrightarrow{v}\cdot \hat{a})$

$\overrightarrow{v}.\hat{a}=\overrightarrow{v}.\frac{\overrightarrow{a}}{|a|}=\overrightarrow{v}.\frac{(\hat{i}+\hat{j}+\hat{k})}{\sqrt{1+1+1}}$

$\Rightarrow \overrightarrow{v}.\hat{a}=\frac{(6\hat{i}+2\hat{j}-2\hat{k}).(\hat{i}+\hat{j}+\hat{k})}{\sqrt{3}}$

$\Rightarrow \overrightarrow{v}.\hat{a}=\frac{6+2-2}{\sqrt{3}}=2\sqrt{3}$

$\Rightarrow$ component $=\left(\overrightarrow{v}\hat{j}\right)\hat{j}=2\sqrt{3}\frac{(\hat{i}+\hat{j}+\hat{k})}{\sqrt{3}}$

$component=2\hat{i}+2\hat{i}+2\hat{k}$