# Relationship between three planes

#### mathmari

##### Well-known member
MHB Site Helper
Hey!!

Let an arbitrary linear system of $3$ equations and $3$ variables be given. There are $4$ cases how the planes can be related.
Describe these $4$ cases graphically and describe the set of solutions in each case.

I have done the following:

If the three equations are linearly independent, then the system has a single solution.
In this case the three planes described by the equations intersect, since they are neither parallel nor identical.
The intersection of these three planes is a point in space.

If two of the three equations are linearly independent, then the system has a set of solution with one free parameter.
In this case the two independent planes described by the equations intersect, since they are neither parallel nor identical. The third one is either parallel or identical to one of the other ones.
The intersection of these two planes is a line in space.

If all the three equations are linearly dependent, then the system has either a set of solution with two free parameters or no solution (empty set of solutions).
In this case the three planes described by the equations are either identical or parallel.
The intersection of these three planes is either a plane (if the planes are identical) or there is no intersection (if the planes are parallel).

Are the four cases correct and complete? Could we improve something?

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Hey!!

Let an arbitrary linear system of $3$ equations and $3$ variables be given. There are $4$ cases how the planes can be related.
Describe these $4$ cases graphically and describe the set of solutions in each case.

I have done the following:

If the three equations are linearly independent, then the system has a single solution.
In this case the three planes described by the equations intersect, since they are neither parallel nor identical.
The intersection of these three planes is a point in space.
Hey mathmari !!

The question seems to ask how the planes can be related rather than whether the equations are linearly independent or not. Doesn't it?

For this first case that is that we have 3 planes that intersect in a point.
The solution is indeed a single point in space.

If two of the three equations are linearly independent, then the system has a set of solution with one free parameter.
In this case the two independent planes described by the equations intersect, since they are neither parallel nor identical. The third one is either parallel or identical to one of the other ones.
The intersection of these two planes is a line in space.
The second case is that we have 3 planes that have exactly one directional vector in common.
Or put otherwise, their normal vectors span a plane.

The solution can indeed be a line.
But isn't it also possible that each combination of 2 planes intersect in different lines?
Or that 2 of the 3 planes are parallel?
In other words, isn't there another possibility for the solution?

If all the three equations are linearly dependent, then the system has either a set of solution with two free parameters or no solution (empty set of solutions).
In this case the three planes described by the equations are either identical or parallel.
The intersection of these three planes is either a plane (if the planes are identical) or there is no intersection (if the planes are parallel).
Third case would be that we have 2 directional vectors in common.
Put otherwise, that their normal vectors are all in the same direction.

That isn't exactly that the three equations are linearly dependent, are they?
Isn't it that each pair of the three equations are linearly dependent instead?

The solution is indeed either a plane (all 3 planes identical), or no solution at all (at least 2 planes are parallel and distinct).

Are the four cases correct and complete? Could we improve something?
Shouldn't there be a fourth case?
Or do we consider the 4 different types of solutions the 4 cases?

There is another case though.
Can't we have a solution that includes all points in 3 dimensional space?

#### mathmari

##### Well-known member
MHB Site Helper
For this first case that is that we have 3 planes that intersect in a point.
The solution is indeed a single point in space.
So case 1 is that all three planes intersect in one point, and then the solution is the intersection point.

The second case is that we have 3 planes that have exactly one directional vector in common.
Or put otherwise, their normal vectors span a plane.

The solution can indeed be a line.
But isn't it also possible that each combination of 2 planes intersect in different lines?
Or that 2 of the 3 planes are parallel?
In other words, isn't there another possibility for the solution?
So in case 2 you mean that the three planes intersect in one line or in different line?

Third case would be that we have 2 directional vectors in common.
Put otherwise, that their normal vectors are all in the same direction.

That isn't exactly that the three equations are linearly dependent, are they?
Isn't it that each pair of the three equations are linearly dependent instead?

The solution is indeed either a plane (all 3 planes identical), or no solution at all (at least 2 planes are parallel and distinct).
In case 3, all three planes are either identical and the intersection is the plane itself or at least two planes are parallel and then there is no intersection.

There is another case though.
Can't we have a solution that includes all points in 3 dimensional space?
Could you explain to me case 4 further? I haven't really understood that.

#### Klaas van Aarsen

##### MHB Seeker
Staff member
So in case 2 you mean that the three planes intersect in one line or in different line?
It depends a bit on how we categorize the relationships that the planes can have.
My current interpretation is that for this case the 3 planes have 1 directional vector in common.
That leads to 2 sub cases for the possible solutions.
One where they have a line in common. That is, the intersection of each pair of planes is an identical line.
And one where there is no solution. That is, the pairwise intersections form at least 2 parallel and distinct lines.

Could you explain to me case 4 further? I haven't really understood that.
The equations do not necessarily form planes.
If all coefficients of an equation are 0, we either have an equation that is always true, or an equation that is always false.
In the first sub case all points in space are solutions to a single equation.
In the second sub case there are no solutions at all to the equation.

The 4 cases correspond to the rank of the matrix of the coefficients.
That rank can be any of {0,1,2,3}, making 4 cases. And each case has sub cases for its solutions.

#### mathmari

##### Well-known member
MHB Site Helper
So, do we have the following four cases?

Case 1: There is no intersection. (The system has no solution.)
This happens for the following relationships of the planes:
• The three planes are parallel but not identical.
• Two identical planes are parallel to the third plane.
• Two planes are parallel and the third plane intersects both planes in two parallel lines.
• All three planes intersect in three different lines.

Case 2: One point intersection. (The system has an unique solution.)
This happens for the following relationship of the planes:
• All three planes intersect in one point.

Case 3: Intersection line. (The system has infinitely many solutions.)
This happens for the following relationships of the planes:
• Two planes are identical and the third plane intersects the other two in a line.
• All three planes intersect in a line.

Case 4: Intersection plane. (The system has infinitely many solutions.)
This happens for the following relationship of the planes:
• All three planes are identical.

Do you mean these cases? Or have I misunderstood something?

#### Klaas van Aarsen

##### MHB Seeker
Staff member
You have categorized by solution now and listed the relationships of the planes as sub cases.
Shouldn't it be the other way around?

Additionally it is possible to have an equation like 0x+0y+0z=1, meaning there is no solution.
And we can also have an equation like 0x+0y+0z=0, meaning that any point in space satisfies this equation.