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Planes in the Space
Sheaf of Planes

A sheaf of planes with an axis, r, is a set of planes that contain the line r

 A1x+B1y+C1z+D1=0 A2x+B2y+C2z+D2=0

How to find the equation of a sheaf of planes.

If the line is

 A1x+B1y+C1z+D1=0 A2x+B2y+C2z+D2=0

The equation of a sheaf of planes with axis, r, is:

 λ(A1x+B1y+C1z+D1)+μ(A2x+B2y+C2z+D2)=0

Dividing by λ and making k=μ/λ, the equation becomes:

(A1x+B1y+C1z+D1)+k(A2x+B2y+C2z+D2)=0

Find the equation of the plane that passes through the point (3,2,-3) and belongs to the sheaf of planes with its axis on the following line:

 2x+3y-z-9=0 -x+2y+3z+2=0

Solution:

2x+3y-z-9+k(-x+2y+3z+2)=0
6+6+3-9+k(-3+4-9+2)=0, that is k=1

Therefore 2x+3y-z-9+(-x+2y+3z+2)=0
The equation of the plane is:
x+5y+2z-7=0