Question

1.Find the projection of the vector i-j on the vector i+j.
2.Find the projection of the vector $2\hat{i}+3\hat{j}+2\hat{k}$ on the vector
$\hat{i}+2\hat{j}+\hat{k}.$.
3.Find the projection of $\overrightarrow{a}=\hat{i}+3\hat{j}+\hat{k}along\overrightarrow{b}=2\hat{i}-3\hat{j}+6\hat{k}.$

Answer 1

$1)$ Projection of vector $\overrightarrow{a}on\overrightarrow{b}$ is given by $\frac{\overrightarrow{a}.\overrightarrow{b}}{\left|\overrightarrow{b}\right|}$

$\frac{(\hat{i}-\hat{j}).(\hat{i}-\hat{j})}{|\hat{i}+\hat{j}|}=\frac{1-1}{\sqrt{2}}=0$

$2)\frac{(2\hat{i}-3\hat{j}+2\hat{k}).(\hat{i}+2\hat{j}+\hat{k})}{|\hat{i}+2\hat{j}+\hat{k}|}=\frac{2+6+2}{\sqrt{1+4+1}}=\frac{10}{\sqrt{16}}$

$3)\frac{\overrightarrow{a}.\overrightarrow{b}}{\left|\overrightarrow{b}\right|}=\frac{(\hat{i}+3\hat{j}+2\hat{k}).(2\hat{i}-3\hat{j}+6\hat{k})}{|2\hat{i}-3\hat{j}+\widehat{6k}|}=\frac{2-9+6}{\sqrt{4+9+36}}=\frac{-1}{\sqrt{49}}=\frac{-1}{7}$