Let a unit vector $\hat{\mathbf{u}}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}$ make angles $\dfrac{\pi}{2},\ \dfrac{\pi}{3}$ and $\dfrac{2\pi}{3}$ with the vectors $\dfrac{1}{\sqrt{2}}\mathbf{i}+\dfrac{1}{\sqrt{2}}\mathbf{k},\ \dfrac{1}{\sqrt{2}}\mathbf{j}+\dfrac{1}{\sqrt{2}}\mathbf{k},\ \dfrac{1}{\sqrt{2}}\mathbf{i}+\dfrac{1}{\sqrt{2}}\mathbf{j}$ respectively. If $\vec{\mathbf{v}}=\dfrac{1}{\sqrt{2}}\mathbf{i}+\dfrac{1}{\sqrt{2}}\mathbf{j}+\dfrac{1}{\sqrt{2}}\mathbf{k}$, then $|\hat{\mathbf{u}}-\vec{\mathbf{v}}|^{2}$ is equal to: