Question

Find the projection of the vector $\overrightarrow{a}$ and $\overrightarrow{b}$, where $\overrightarrow{a}=3\hat{i}-2\hat{j}+\hat{k}$ and $\overrightarrow{b}=\hat{i}-2\hat{j}-3\hat{k}$.

Answer 1

Projection of $\overrightarrow{a}$ on $\overrightarrow{b}$ $=$ $\frac{\overrightarrow{a.}\overrightarrow{b}}{b}$

Given,

$\overrightarrow{a}=3\hat{i}-2\hat{j}+\hat{k}$ and $\overrightarrow{b}=\hat{i}-2\hat{j}-3\hat{k}$

$\overrightarrow{a}.\overrightarrow{b}$ $=$ $3\left(1\right)-2(-2)+1(-3)$

$=$ $3+4-3$

$=$ $4$

$|\overrightarrow{b}|$$=$ $\sqrt{(1{)}^2+(-2{)}^2+{(-3)}^2}$

$=$ $\sqrt{1+4+9}$

$=$ $\sqrt{14}$

Projection of $\overrightarrow{a}$ on $\overrightarrow{b}$ $=$ $\frac{\overrightarrow{a.}\overrightarrow{b}}{b}$

$=$ $\frac{4}{\sqrt{14}}$