Example 2 The line $l_{1}$ passes through the point $P(3,2)$, and its inclination angle is $\arctan \frac{3}{4}$. If $l_{1}$ intersects the line $l_{2}: x-2 y+11=0$ at point $Q$, find $|P Q|$.
The parametric equation of $l_{1}$ is $\left\{\begin{array}{l}x=3+\frac{4}{5} t, \\ y=2+\frac{3}{5} t\end{array}\right.$. $t$ is the parameter.
Substituting it into $x-2 y+11=0$, we get $3+\frac{4}{5} t-2\left(2+\frac{3}{5} t\right)+11=0 . t=25$, hence $|P Q|=|t|=25$.