Compose the equation of the plane passing through the midpoint of the segment with endpoints at points $P(-1 ; 2 ; 5)$ and $Q(3 ;-4 ; 1)$, perpendicular to the line passing through points $A(0 ;-2 ;-1)$ and $B(3 ; 2 ;-1)$.
The coordinates of the midpoint $M$ of the segment $P Q$ are equal to the arithmetic means of the coordinates of its ends, i.e., ( $1 ; -1 ; 3$ ).

The desired plane is perpendicular to the vector $\overrightarrow{A B}=(3-0 ; 2-(-2) ;-1-(-1))=(3 ; 4 ; 0)$, hence, its equation has the form

$$
3(x-1)+4(y+1)+0(z-3)=0, \text { or } 3 x+4 y+1=0 \text {. }
$$

## Answer

$3 x+4 y+1=0$.