Vector rotation around Z-axis

You can solve this problem by using the $3 \times 3$ rotation matrix about Z-axis by angle $\theta = 180^\circ$ :

$\begin{aligned} R_{z,\theta} &= \left(\begin{array}{ccc} \cos(\theta) & -\sin(\theta) & 0\\ \sin(\theta) &\cos(\theta) &0 \\ 0 & 0 &1 \\ \end{array}\right) = \left(\begin{array}{ccc} -1 & 0 & 0\\ 0 &-1 &0 \\ 0 & 0 &1\\ \end{array}\right) \\ \\ \\ \text{Rotated vector } &= {[R_{z,180}]}. \vec V = \left(\begin{array}{ccc} -1 & 0 & 0\\ 0 &-1 &0 \\ 0 & 0 &1\\ \end{array}\right) . \left(\begin{array}{c} 2\\ 5 \\ 3\\ \end{array}\right) = \left(\begin{array}{c} -2\\ -3 \\ 5\\ \end{array}\right) \end{aligned}$