Question

Find the projections of the vector $\hat{i}-\hat{j}$ on the vector $\hat{i}-\hat{j}$

Answer 1

Let $\overrightarrow{a}=\hat{i}-\hat{j},$

$\overrightarrow{b}=\hat{i}-\hat{j}$

Scalar projection of $\overrightarrow{a}$ on $\overrightarrow{b}$

$=\frac{\overrightarrow{a}.\overrightarrow{b}}{|\overrightarrow{a}|}$

$=\frac{(\hat{i}-\hat{j}).(\hat{i}-\hat{j})}{\sqrt{{\left(1\right)}^2+{(-1)}^2}}$

$=\frac{1^2+{(-1)}^2}{\sqrt{2}}$

$=\frac{2}{\sqrt{2}}$

$=\sqrt{2}$

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

$=\frac{\overrightarrow{a}.\overrightarrow{b}}{|\overrightarrow{a}{|}^2}.(\hat{i}+\hat{j}+\hat{k})$

$=\frac{(\hat{i}-\hat{j}).(\hat{i}-\hat{j})}{{\left(\sqrt{{\left(1\right)}^2+{(-1)}^2}\right)}^2}.(\hat{i}+\hat{j}+\hat{k})$

$=\frac{1^2+{(-1)}^2}{{\left(\sqrt{2}\right)}^2}.(\hat{i}+\hat{j}+\hat{k})$

$=\frac{2}{2}.(\hat{i}+\hat{j}+\hat{k})$

$=(\hat{i}+\hat{j}+\hat{k})$