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