Line-Plane Intersection

1. Line Defined by Two Planes

The line is Line Equation and the plane isPlane Equation.

Form a system with the equations and calculate the ranks.

System of Equations

r = rank of the coefficient matrix.

r'= rank of the augmented matrix.

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

Point Intersection

r = 3       r' = 3

Point Intersection

No Intersection

r = 2       r' = 3

No Intersection

Line Intersection

r = 2       r' = 2

Line Intersection


Examples

State the relationship between the line and the plane:

1. Line and Plane Intersection Example

Form the system of equations.

System of Equations

Find the rank of the coefficient matrix.

Coefficient Matrix

Determine the rank of the augmented matrix.

Augmented Matrix

Compare the ranks.

Point intersection.


2. Line and Plane Intersection Example

System of Equations

Coefficient Matrix

Augmented Matrix

The line and plane are parallel.


2. The Line Is Defined by a Point and a Vector

The line is defined by Point A, the vector, Vector U, and the plane by the normal vector, Normal Vector.

Intersection producto escalar A
Line intersection = 0 pertenece π
No intersection = 0 no pertenece π
Point intersection ≠ 0