Example 7 Let $M$ be a moving point on the unit circle $x^{2}+y^{2}=1$, and $N$ be a vertex of an equilateral triangle with fixed point $A(3,0)$ and $M$. Assume that $M \rightarrow N \rightarrow A \rightarrow M$ forms a counterclockwise direction. Find the equation of the trajectory of point $N$ as point $M$ moves.
【Analysis】This problem is relatively difficult to solve using the analytical method. The following solution uses the geometric meaning of complex number multiplication.

Let the complex numbers corresponding to $M$, $N$, and $A$ be $x^{\prime}+\mathrm{i} y^{\prime}$, $x+\mathrm{i} y$, and $3$ respectively.

Then the vector $\overrightarrow{A M}$ can be obtained by rotating $\overrightarrow{A N}$ counterclockwise by $300^{\circ}$ around point $A$. This can be represented by complex number operations as
$$
\begin{array}{l}
\overrightarrow{A M}=\left(\cos \frac{5 \pi}{3}+\mathrm{i} \sin \frac{5 \pi}{3}\right) \overrightarrow{A N} \\
\Rightarrow \overrightarrow{O M}-\overrightarrow{O A} \\
\quad=\left(\cos \frac{5 \pi}{3}+\mathrm{i} \sin \frac{5 \pi}{3}\right)(\overrightarrow{O N}-\overrightarrow{O A}) \\
\Rightarrow x^{\prime}+\mathrm{i} y^{\prime}-3=\frac{1-\sqrt{3} \mathrm{i}}{2}(x+\mathrm{i} y-3) \\
\Rightarrow x^{\prime}=\frac{x+\sqrt{3} y+3}{2}, y^{\prime}=\frac{y-\sqrt{3} x+3 \sqrt{3}}{2} . \\
\text { Also, } x^{\prime 2}+y^{\prime 2}=1 \text {, thus, } \\
\left(x-\frac{3}{2}\right)^{2}+\left(y+\frac{3 \sqrt{3}}{2}\right)^{2}=1 .
\end{array}
$$