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Maths Exercices

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
: 4x+8y+3z=4 and the line -x-2y=2
x+2y+z=2
  1. Point Intersection
  2. No Intersection
  3. Line Intersection