To study the intersection of three planes, form a system with the equations of the planes and calculate the ranks.

r = rank of the coefficient matrix.

r'= rank of the augmented matrix.

The relationship between three planes presents can be described as follows:

1. Intersecting at a Point

r=3, r'=3

2.1 Each Plane Cuts the Other Two in a Line.

r = 2, r' = 3

The three planes form a prismatic surface.

2.2 Two Parallel Planes and the Other Cuts Each in a Line

r = 2, r' = 3

Two rows of the coefficient matrix are proportional.

3.1 Three Planes Intersecting in a Line

r = 2, r' = 2

3.2 Two Coincident Planes and the Other Intersecting Them in a Line

r = 2, r' = 2

Two rows of the augmented matrix are proportional.

4.1 Three Parallel Planes

r = 1, r' = 2

4.2 Two Coincident Planes and the Other Parallel

r = 1, r' = 2

Two rows of the augmented matrix are proportional.

5. Three Coincident Planes

r = 1, r' = 1

State the relationship between the three planes.


Each plane cuts the other two in a line and they form a prismatic surface.


Each plan intersects at a point.


The second and third planes are coincident and the first is cuting them, therefore the three planes intersect in a line.


The first and second are coincident and the third is parallel to them.

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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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