Question

Find the projection of the vector $\overrightarrow{a}=4\hat{i}-2\hat{j}+\hat{k}$ on the vector $\overrightarrow{b}=3\hat{i}+6\hat{j}+2\hat{k}$. Also find component of $\overrightarrow{a}$ along $\overrightarrow{b}$ and perpendicular to $\overrightarrow{b}$.

Answer 1

$\overrightarrow{a}=4\hat{i}-2\hat{j}+\hat{k}$

$\overrightarrow{b}=3\hat{i}+6\hat{j}+2\hat{k}$

$\left|\overrightarrow{b}\right|=\sqrt{9+36+4}=7$

$\hat{b}=\frac{\overrightarrow{b}}{|\overrightarrow{b}|}=\frac{3}{7}\hat{i}+\frac{6}{7}\hat{j}+\frac{2}{7}\hat{k}$

Projection of $\overrightarrow{a}$ on $\overrightarrow{b}=\overrightarrow{a}.\hat{b}=\left(4\hat{i}-2\hat{j}+\hat{k}\right).\left(\frac{3}{7}\hat{i}+\frac{6}{7}\hat{j}+\frac{2}{7}\hat{k}\right)=\frac{2}{7}$

Component of vector $\overrightarrow{a}$ on $\overrightarrow{b}=\frac{2}{7}\hat{b}=\frac{2}{7}\left(\frac{3}{7}\hat{i}+\frac{6}{7}\hat{j}+\frac{2}{7}\hat{k}\right)=\frac{3}{49}\hat{i}+\frac{6}{49}\hat{j}+\frac{2}{49}\hat{k}$

Component of $\overrightarrow{a}$ along perpendicular to $\overrightarrow{b}=\overrightarrow{a}-\frac{2}{7}\hat{b}$

$=\frac{190}{49}\hat{i}-\frac{110}{49}\hat{j}+\frac{45}{49}\hat{k}$