Let a unit vector \(\hat{u}=x \hat{i}+y \hat{j}+z \hat{k}\) make angles \(\frac{\pi}{2}, \frac{\pi}{3}\) and \(\frac{2 \…

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

[JEE Main 2024, 29 Jan (Shift 2)]