Intersection Between Two Lines

1. Lines Defined by Two Planes

Line Defined by Two Planes

Line Defined by Two Planes

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

System of Equations

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

Skew Lines

Intersecting Lines

r = 3       r' = 3

Intersecting Lines

Parallel Lines

r = 2       r' = 3

Parallel Lines

Coincident Lines

r = 2       r' = 2

Coincident Lines


Examples

State the relationship between the following lines:

1. Line Relationship

Form a 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.

They are skew lines.


2. Line Relationship

System of Equations

Coefficient Matrix

Augmented Matrix

They are intersecting lines.


2. Lines Defined by a Point and a Vector

If the line, r, is determined by Line Defined by a Point and Vector and Line Defined by a Point and Vector and the line, s, is determined by Line Defined by a Point and Vector and Vector V, the intersection of r and s is given by the position of Vectors.

If Linearly Dependent Vectors, there are two possibilities:

1. Coincident lines if Coincident Lines.

2.Parallel lines if Parallel Lines.

If Linearly Dependent Vectors, there are two other possibilities:

3. Intersecting lines if Intersecting Lines.


4. Skew lines if Skew Lines.