Let a unit vector which makes an angle of $60^\circ$ with $\,2\hat i+2\hat j-\hat k\,$ and an angle of $45^\circ$ with $\,\hat i-\hat k\,$ be $\vec C$. Then $\displaystyle \vec C+\Big(-\tfrac12\,\hat i+\tfrac{1}{3\sqrt2}\,\hat j-\tfrac{\sqrt2}{3}\,\hat k\Big)$ is: