Let a unit vector $\hat{\mathbf u}=x\hat i+y\hat j+z\hat k$ make angles $\dfrac{\pi}{2},\ \dfrac{\pi}{3}$ and $\dfrac{2\pi}{3}$ with the vectors $\dfrac{1}{\sqrt2}\hat i+\dfrac{1}{\sqrt2}\hat k$, $\dfrac{1}{\sqrt2}\hat j+\dfrac{1}{\sqrt2}\hat k$ and $\dfrac{1}{\sqrt2}\hat i+\dfrac{1}{\sqrt2}\hat j$ respectively. If $\vec v=\dfrac{1}{\sqrt2}\hat i+\dfrac{1}{\sqrt2}\hat j+\dfrac{1}{\sqrt2}\hat k$, then $|\hat{\mathbf u}-\vec v|^{2}$ is equal to: