5.1.1 * Given complex numbers $z_{1}, z_{2}$ satisfy the following conditions:
$$
\left|z_{1}\right|=2,\left|z_{2}\right|=3,3 z_{1}-2 z_{2}=2-\mathrm{i} \text {. }
$$

Find the value of $z_{1} z_{2}$.
From the conditions, we know that $z_{1} \overline{z_{1}}=4, z_{2} \overline{z_{2}}=9$, thus
$$
3 z_{1}-2 z_{2}=\frac{1}{3} z_{1} z_{2} \overline{z_{2}}-\frac{1}{2} z_{2} z_{1} \overline{z_{1}}=\frac{1}{6} z_{1} z_{2}\left(2 \overline{z_{2}}-3 \overline{z_{1}}\right) \text {. }
$$

Therefore, we have
$$
\begin{aligned}
z_{1} z_{2} & =\frac{6\left(3 z_{1}-2 z_{2}\right)}{2 \overline{z_{2}}-3 \overline{z_{1}}}=-\frac{6\left(3 z_{1}-2 z_{2}\right)}{\overline{3 z_{1}-2 z_{2}}}=-\frac{6(2-\mathrm{i})}{2+\mathrm{i}} \\
& =-\frac{6}{5}(2-\mathrm{i})^{2}=-\frac{18}{5}+\frac{24}{5} \mathrm{i}
\end{aligned}
$$

Hence, the value of $z_{1} z_{2}$ is $-\frac{18}{5}+\frac{24}{5} \mathrm{i}$.