Example 4. Determine the distance from the point $P(-7,-13,10)$ to the line $l: \frac{x-1}{-2}=\frac{y+2}{1}=\frac{z}{0}$.
Solution. 1) Through $P$ we draw a plane $\alpha$ perpendicular to $l$, taking $\vec{N}=\vec{l}=\{-2,1,0\}$. We obtain $(\alpha):-2(x+7)+(y+13)=0$, i.e., $2 x-y+1=0$.

2) We find the coordinates of the point $O$ of intersection of $l$ and $\alpha$. Substituting the expressions $x=1-2 t, y=-2+t, z=0$ into the equation of the plane: $2(1-2 t)-(t-2)+1=0$. First, we find $t=1$, then $x=-1$, $y=-1, z=0$, i.e., $O(-1,-1,0)$.
3) The desired distance is

$$
O P=\sqrt{(-7+1)^{2}+(-13-1)^{2}+10^{2}}=2 \sqrt{83}
$$

Answer. $d=2 \sqrt{83}$.