51. Given the equations of a plane and a line:

$$
A x+B y+C z+D=0 \text { and } \frac{x-x_{1}}{m}=\frac{y-y_{1}}{n}=\frac{z-z_{1}}{p}
$$

Find the conditions for the parallelism and perpendicularity of the line and the plane
51. For the parallelism of a line and a plane, it is necessary and sufficient that the direction vector of the line be perpendicular to the normal vector of the plane: a line perpendicular to the perpendicular of the plane will be parallel to this plane. Therefore, the condition for parallelism can be written as:

$$
A m+B n+C p=0
$$

In the case of perpendicularity of the line and the plane, the same vectors must be collinear, i.e., their coordinates are proportional to each other:

$$
\frac{A}{m}=\frac{B}{n}=\frac{C}{p}
$$