LinePlane 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

A_{1}x+B_{1}y+C_{1}z+D_{1}=0

A_{2}x+B_{2}y+C_{2}z+D_{2}=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.





r=3 





r'=3 
Point Intersection.
State the relationship between the line:
and the plane
Solution:
Form the system of equations and calculate the ranks.




=0 
r=2 





r'=3 
The line and plane are parallel. There is no intersection.
State the relationship between the plane