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