# Line-Plane Intersection

### 1. Line Defined by Two Planes

The line is 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 follows:

#### Point Intersection

r = 3 r' = 3

#### No Intersection

r = 2 r' = 3

#### Line Intersection

r = 2 r' = 2

#### Examples

State the relationship between the line and the plane:

1.

Form the system of equations.

Find the rank of the coefficient matrix.

Determine the rank of the augmented matrix.

Compare the ranks.

Point intersection.

2.

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, , and the plane by the normal vector, .

Intersection | A | |
---|---|---|

Line intersection | = 0 | ∈ π |

No intersection | = 0 | ∉ π |

Point intersection | ≠ 0 |

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