Example 5 Given complex numbers $z_{1}, z_{2}$ satisfy $\left|z_{1}\right|=2,\left|z_{2}\right|=3$. If the angle between the vectors they correspond to is $60^{\circ}$, find $\frac{\left|z_{1}+z_{2}\right|}{\left|z_{1}-z_{2}\right|}$.
(2002 National High School League Question)
As shown in Figure $7-3$, by the geometric meaning of the addition and subtraction of complex numbers and the cosine theorem, we easily get:
$$
\left|z_{1}+z_{2}\right|=\sqrt{19},\left|z_{1}-z_{2}\right|=\sqrt{7} \text {, so } \frac{\left|z_{1}+z_{2}\right|}{\left|z_{1}-z_{2}\right|}=\frac{\sqrt{19}}{\sqrt{7}}=\frac{\sqrt{133}}{7} \text {. }
$$