Question

Write the three times of the projection of the vector $\overrightarrow{a}=2\hat{i}-\hat{j}+\hat{k}$ on the vector $\overrightarrow{b}=\hat{i}+2\hat{j}+2\hat{k}$

Answer 1

Given vectors are $\overrightarrow{a}=2\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{b}=\hat{i}+2\hat{j}+2\hat{k}$
Projection of $\overrightarrow{a}$ on the vector $\overrightarrow{b}$ is $\frac{\overrightarrow{a}.\overrightarrow{b}}{\left|\overrightarrow{b}\right|}$
let p be the projection of $\overrightarrow{a}$ on the vector $\overrightarrow{b}$
$\overrightarrow{a}.\overrightarrow{b}=(2\hat{i}-\hat{j}+\hat{k}).(\hat{i}+2\hat{j}+2\hat{k})=2$
$\Rightarrow p=\frac{\overrightarrow{a}.\overrightarrow{b}}{\left|\overrightarrow{b}\right|}=\frac{2}{3}$
$\overrightarrow{p}=p\hat{b}=\frac{2}{9}(\hat{i}+2\hat{j}+2\hat{k})$

$\therefore$ $\text{three times the projection =3p=2}$