Example 5. Find the corresponding complex number of the vector $\overrightarrow{O M}_{\text {! }}$ obtained by rotating the complex number $3+4 \mathrm{i}$ counterclockwise around the origin by $30^{\circ}$.
The complex number corresponding to the vector $\mathrm{OM}_{1}{ }^{\prime}$ should be:
$$
\begin{aligned}
& (3+4 \mathrm{i})\left(\cos 30^{\circ}+\mathrm{i} \sin 30^{\circ}\right)=(3+4 \mathrm{i})\left(\frac{\sqrt{3}}{2}+\frac{1}{2} \mathrm{i}\right) \\
= & \left(\frac{3 \sqrt{3}}{2}-2\right)+\left(2 \sqrt{3}+\frac{3}{2}\right) \mathrm{i} \\
= & \frac{3 \sqrt{3}-4}{2}+\frac{4 \sqrt{3}+3}{2} \mathrm{i}
\end{aligned}
$$