Problem 8. Form the equation of the plane $\alpha$, passing through the midpoint $M$ of the segment $A D$ and perpendicular to the line $C B$, if $A(-1 ; 2 ;-3), D(-5 ; 6$; $-1), C(-3 ; 10 ;-5), B(3 ; 4 ; 1)$.
Solution. Obviously, $M(-3 ; 4 ;-2)$. Let $P(x ; y ; z)$ be an arbitrary point on the plane $\alpha$. Then $\overline{M P} \cdot \overline{C B}=0$, or $(x+3 ; y-4 ; z+2) \cdot(6 ;-6 ; 6)=0$, or $6(x+3)-6(y-4)+6(z+2)=0$.

Thus, the equation of the plane $\alpha$ is: $z-y+x+9=0$.