50. Given equations of two planes:

$$
A_{1} x+B_{1} y+C_{1} z+D_{1}=0 \text { and } A_{2} x+B_{2} y+C_{2} z+D_{2}=0
$$

Find the conditions for the parallelism and perpendicularity of these planes.
50. The coefficients of the variables in the equations of planes are the coordinates of vectors perpendicular to these planes, i.e., normal vectors. Therefore, for the parallelism of planes, the necessary and sufficient condition is the proportionality of these coefficients to each other:

$$
\frac{A_{1}}{A_{2}}=\frac{B_{1}}{B_{2}}=\frac{C_{1}}{C_{2}}
$$

For the perpendicularity of planes, the condition is the scalar product of the normal vectors being zero:

$$
A_{1} A_{2}+B_{1} B_{2}+C_{1} C_{2}=0
$$