Question #40883

A point $P(\sqrt{3}, 1, 2)$ in 3D space is projected onto the $xy$-plane to obtain the point $P'$. The coordinates of $P'$ are represented by the complex number $Z_0 = x + iy$. $Z_0$ is rotated about the origin in the complex plane by an angle $\theta = \frac{1}{2} \arg(W)$ to obtain the complex number $Z_1$, where $W = \frac{(-1 + i\sqrt{3})^5}{(1 + i\sqrt{3})^2}$ and $\arg(W)$ denotes the principal argument of $W$ such that $\arg(W) \in (-\pi, \pi]$. A new point $Q$ in 3D space is formed such that its projection on the $xy$-plane corresponds to $Z_1$ and its $z$-coordinate is the same as that of $P$. If $O$ is the origin, find the value of $|\vec{OP} \times \vec{OQ}|^2$.