The angle between straight lines whose direction cosines are $\left(\frac{1}{2},-\frac{1}{2}, \frac{1}{\sqrt{2}}\right)$ and $\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}\right)$ is

$\cos ^{-1}\left(-\frac{1}{\sqrt{6}}\right)$

$\theta=\cos ^{-1}\left(\frac{\frac{1}{2 \sqrt{3}}-\frac{1}{2 \sqrt{3}}-\frac{1}{\sqrt{6}}}{\sqrt{\frac{1}{4}+\frac{1}{4}+\frac{1}{2}} \sqrt{\frac{1}{3}+\frac{1}{3}+\frac{1}{3}}}\right)=\cos ^{-1}\left(\frac{-1}{\sqrt{6}}\right)$