Example 5. For what values of $B$ and $C$ is the line $\frac{x-3}{1}=\frac{y-2}{B}=$ $=\frac{z+2}{C}$ perpendicular to the plane $3 x-2 y+5 z=0$?
Solution. The condition of perpendicularity of a line and a plane is equivalent to the condition of parallelism of their vectors $\vec{l}=\{1, B, C\}$ and $\vec{N}=$ $=\{3,-2,5\}$. The corresponding coordinates of these vectors must be proportional ( $\$ 3$ ): $\frac{1}{3}=\frac{B}{-2}=\frac{C}{5}$. From this, $B=-\frac{2}{3}, C=\frac{5}{3}$.