The angle \(\theta\) between the vectors \(\vec{a}=\hat{i}+\hat{j}-\hat{k}\) and \(\vec{b}=\hat{i}-\hat{j}+\hat{k}\) is

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

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

\(\begin{aligned}\cos \theta &=\frac{(\hat{i}+\hat{j}-\hat{k})\cdot (\hat{i}-\hat{j}+\hat{k})}{\sqrt{3}\cdot \sqrt{3}}\\ &=-\frac{1}{3}\\ \theta &=\cos^{-1}\left(-\frac{1}{3}\right)\end{aligned}\)