Question

The vector component of the vector $\hat{i}+\hat{j}+\hat{k}$ along the vector $2\hat{i}-\hat{j}+2\hat{k}$ is

• A. $\frac{1}{2}(\hat{i}+\hat{j}+\hat{k})$
• B. $2\hat{i}-\hat{j}+2\hat{k}$
• C. $\frac{1}{2}(2\hat{i}-\hat{j}+2\hat{k})$
• D. $\frac{1}{3}(2\hat{i}-\hat{j}+2\hat{k})$

Answer 1

The correct option is D $\frac{1}{3}(2\hat{i}-\hat{j}+2\hat{k})$

The vector component of $\overrightarrow{a}$ along $\overrightarrow{b}$ is $\left(\frac{\overrightarrow{a}.\overrightarrow{b}}{\overrightarrow{b}.\overrightarrow{b}}\right).\overrightarrow{b}$
$\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}$
$\overrightarrow{b}=2\hat{i}-\hat{j}+2\hat{k}$
$\Rightarrow \left(\frac{\overrightarrow{a}.\overrightarrow{b}}{\overrightarrow{b}.\overrightarrow{b}}\right)$
$=\frac{(\hat{i}+\hat{j}+\hat{k}).(2\hat{i}-\hat{j}+2\hat{k})}{(2\hat{i}-\hat{j}+2\hat{k}).(2\hat{i}-\hat{j}+2\hat{k})}$
$=\frac{2-1+2}{4+1+4}=\frac{1}{3}$ on simplification
$\therefore$ the component is $\frac{1}{3}\overrightarrow{b}=\frac{1}{3}(2\hat{i}-\hat{j}+2\hat{k})$