Planes in the Space

# Line-Plane Intersection

In analytic geometry, the intersection of a line and a plane can be the empty set, a point or a line:

 No Intersection Point Intersection Line Intersection

How to find the relationship between a line and a plane.

If the line is

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

and the plane is

Form a system with the equations and calculate the ranks.

r = rank of the coefficient matrix
r'= rank of the augmented matrix

The relationship between the line and the plane can be described as follow:

 Case 1. Point intersection r=3 and r'=3 Case 2. No Intersection r=2 and r'=3 Case 3. Line Intersection r=2 and r'=2

State the relationship between the line:

and the plane

Solution:

Form the system of equations and calculate the ranks.

 1 -2 0 0 1 1 1 -2 3
r=3

 1 -2 0 0 1 1 1 -2 3
r'=3

Point Intersection.

State the relationship between the line:

and the plane

Solution:

Form the system of equations and calculate the ranks.

 1 -5 0 0 1 -1 -1 3 2
=0 r=2

 1 -5 1 0 1 2 -1 3 -5
r'=3

The line and plane are parallel. There is no intersection.

State the relationship between the plane
 : 2x+7y+2z=-4 and the line -2x-2z=-6 +3y=-7
1. Point Intersection
2. No Intersection
3. Line Intersection