Let a unit vector u =x i +y j +z k make angles (π)/2, (π)/3 and (2 π)/3 with the vectors 1/(√ 2) i +1/(√ 2) k, 1/(√ 2) j +1/(√ 2) k and 1/(√ 2) i…

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