Find angle '$\theta$' between the vectors $\vec{a} = \hat{i} + \hat{j} - \hat{k}$ and $\vec{b} = \hat{i} - \hat{j} + \hat{k}$.

$\cos^{-1} \left( -\frac{1}{3} \right)$

The correct answer is Option (2) → $\cos^{-1} \left( -\frac{1}{3} \right)$ ##

The angle $\theta$ between two vectors $\vec{a}$ and $\vec{b}$ is given by

$\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}$

Now $\vec{a} \cdot \vec{b} = (\hat{i} + \hat{j} - \hat{k}) \cdot (\hat{i} - \hat{j} + \hat{k}) = 1 - 1 - 1 = -1$.

Therefore, we have $\cos \theta = \frac{-1}{3}$

hence the required angle is $\theta = \cos^{-1}\left( -\frac{1}{3} \right)$