Explanation
unit vector u^=xi^+yj^+zk^ makes 2π,3π \& 32π
\begin{aligned}
& \text{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} & \\
& \frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{j}\text{respectively.} \\
& \therefore\frac{x}{\sqrt{2}}+\frac{z}{\sqrt{2}}=\left|\hat{u}\right|\sqrt{\frac{1}{2}+\frac{1}{2}}.\cos\sqrt{\frac{\pi}{2}} \\
& & & =0 \\
& & & x+z=0 \\
& & & \sqrt{\frac{y}{2}}+\frac{z}{\sqrt{2}}=\left|\hat{u}\right|\sqrt{\frac{1}{2}+\frac{1}{2}}\cos\frac{\pi}{3} \\
& & & \frac{y+z}{\sqrt{2}}=\frac{1}{2} \\
& & & y+z=\frac{1}{\sqrt{2}}....(\mathrm{ii}) \\
& & & \frac{x}{\sqrt{2}}+\frac{y}{\sqrt{2}}=\left|\hat{a}\right|\sqrt{\frac{1}{2}+\frac{1}{2}}\cos\frac{2\pi}{3} \\
& & & \frac{x+y}{\sqrt{2}}=1\times1\times-\frac{1}{2} \\
& & & x+y=-\frac{1}{\sqrt{2}}...(\mathrm{ii})
\end{aligned}
\begin{align*}
&\text{add (i), (ii) \& (iii)} \\
&7x + 2y + 2z = 0 \\
&x + y + z = 0 \\
&y = 0 \\
&x = -\frac{1}{\sqrt{2}} \\
&z = \frac{1}{\sqrt{2}} \\
&\hat{u} = -\frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{k} \\
&|\hat{u} - \text{J}|^2 = \left(\frac{2}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 + 0^2 \\
&= \frac{4}{2} + \frac{1}{2} = \frac{5}{2}
\end{align*}
