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 
.
