# Intersection Between Two Lines

### 1. Lines Defined by Two Planes

Form a system with the equations of lines and calculate the ranks.

r = rank of the coefficient matrix.

r'= rank of the augmented matrix.

The relationship between two lines can be described as follows:

#### Skew Lines

r = 3       r' = 4

#### Intersecting Lines

r = 3       r' = 3

#### Parallel Lines

r = 2       r' = 3

#### Coincident Lines

r = 2       r' = 2

#### Examples

State the relationship between the following lines:

1.

Form a system of equations.

Find the rank of the coefficient matrix.

Determine the rank of the augmented matrix.

Compare the ranks.

They are skew lines.

2.

They are intersecting lines.

### 2. Lines Defined by a Point and a Vector

If the line, r, is determined by and and the line, s, is determined by and , the intersection of r and s is given by the position of .

If , there are two possibilities:

1. Coincident lines if .

2.Parallel lines if .

If , there are two other possibilities:

3. Intersecting lines if .

4. Skew lines if .