Question

If $\overrightarrow{a}=2\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=5\hat{i}-3\hat{j}+\hat{k}$, then orthogonal projection of $\overrightarrow{a}$ on $\overrightarrow{b}$ is -

• A. $3\hat{i}-3\hat{j}+\hat{k}$
• B. $\frac{9(5\hat{i}-3\hat{j}+\hat{k})}{35}$
• C. $8\frac{(5\hat{i}-3\hat{j}+\hat{k})}{35}$
• D. $9(5\hat{i}-3\hat{j}+\hat{k})$

Answer 1

The correct option is C $8\frac{(5\hat{i}-3\hat{j}+\hat{k})}{35}$

Projection of $\overrightarrow{a}$ on $\overrightarrow{b}$ is given by

$\Rightarrow proj=\frac{\overrightarrow{a}.\overrightarrow{b}}{|b|}\times \frac{\overrightarrow{b}}{|b|}$

$\Rightarrow proj=\frac{(2\hat{i}+\hat{j}+\hat{k}).(5\hat{i}-3\hat{j}+\hat{k})}{\sqrt{5^2+3^2+1^2}}\times \frac{5\hat{i}-3\hat{j}+\hat{k}}{\sqrt{5^2+3^2+1^2}}$

$\Rightarrow proj=\frac{10-3+1}{\sqrt{35}}\times \frac{5\hat{i}-3\hat{j}+\hat{k}}{\sqrt{35}}$

$\Rightarrow proj=\frac{8}{35}\times 5\hat{i}-3\hat{j}+\hat{k}$

$\Rightarrow proj=\frac{8(5\hat{i}-3\hat{j}+\hat{k})}{35}$