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

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

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

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

$\overrightarrow{a}.\overrightarrow{b}=1.1+1(-1)+(-1).1=-1$

$\overrightarrow{a}.\overrightarrow{b}=$ $\left|\overrightarrow{a}\right|.\left|\overrightarrow{b}\right|cos\theta$

$-1=\sqrt{3}\ast \sqrt{3}cos\theta$

$⟹cos\theta =-\frac{1}{3}$

$⟹\theta =co{s}^{-1}(-\frac{1}{3})$