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

State the relationship between the following lines:

Example 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.

Example 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 .

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I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.

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