Question

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

• A. $\frac{1}{3}(\hat{i}+4\hat{j}+\hat{k})$
• B. $\frac{1}{4}(2\hat{i}-2\hat{j}-3\hat{k})$
• C. $\frac{1}{2}(2\hat{i}+2\hat{j}-\hat{k})$
• D. $\frac{1}{4}(2\hat{i}+2\hat{j}-\hat{k})$

Answer 1

The correct option is A $\frac{1}{3}(\hat{i}+4\hat{j}+\hat{k})$

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