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 $\vec{c} + \left( -\frac{1}{2}\hat{i} + \frac{1}{3\sqrt{2}}\hat{j} - \frac{\sqrt{2}}{3}\hat{k} \right)$ is:
