Question

Let $\overrightarrow{a}=2\stackrel{\wedge }{i}+3\stackrel{\wedge }{j}+2\stackrel{\wedge }{k}$ and $\overrightarrow{b}=\stackrel{\wedge }{i}+2\stackrel{\wedge }{j}+\stackrel{\wedge }{k}$.Find the component of $\overrightarrow{b}$ on $\overrightarrow{a}$.

Answer 1

Given,

$\begin{array}{c}\overrightarrow{a}=2\hat{i}+3\hat{j}+2\hat{k}\\ \overrightarrow{b}=\hat{i}+2\hat{j}+\hat{k}\end{array}$

Vector projection of $\overrightarrow{b}$ on to $\overrightarrow{a}$

$\begin{array}{c}=\left(\frac{\overrightarrow{a}.\overrightarrow{b}}{\left|\overrightarrow{a}\right|}\right).\frac{\overrightarrow{a}}{\left|\overrightarrow{a}\right|}\\ =\left(\frac{2+6+2}{\sqrt{4+9+4}}\right)\frac{\left(2\hat{i}+3\hat{j}+2\hat{k}\right)}{\sqrt{4+9+4}}\\ =\frac{10}{17}\left(2\hat{i}+3\hat{j}+2\hat{k}\right)\end{array}$