Question

Find the angle between the vectors $\overrightarrow{a}=\hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}$

Answer 1

Given $\overrightarrow{a}=\hat{i}+\hat{j}-\hat{k},\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}$

$\overrightarrow{a}.\overrightarrow{b}=1-1-1=-1$

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

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

Let the angle between the vectors be $\theta$

$⟹\cos \theta =\frac{\overrightarrow{a}.\overrightarrow{b}}{|\overrightarrow{a}||\overrightarrow{b}|}=\frac{-1}{(\sqrt{3}{)}^2}=\frac{-1}{3}$