When working with matrices, and determining their geometric description, how do we know whether they are parallel, the same, or have the planes intersect in either a line or at a single point?

I am referring to both questions with three equations in three variable, and questions with two equations, but three variables.

Thank you in advance!!

Essentially, the geometric interpretation of the solutions (and hence the original planes) depends on the nature of the solutions we had found.

In the 3 equation, 3 variable case, we are considering 3 planes in \( \mathbb{R}^3\). So we have the following:

- There is no solution. When this is the case, we're saying that the 3 planes

**have no common point of intersection**. This could be because they appear to form a triangular prism between them, or because at least 2 out of the 3 planes are parallel to one another.

- There is one unique solution. When this is the case, the planes are essentially intersecting

**at a common point**. An easy example of this is just the x-y, y-z and x-z planes, which only intersect at (0,0,0).

- There are infinite solutions. Infinite solutions are characterised by the number of parameters there are in our solution.

- If there is only one parameter, say \(t\), then the solutions are of the form \( \textbf{x} = a + t\textbf{v} \). This represents a line in \(\mathbb{R}^3\), passing through the point \(A\) and in the direction of the vector \( \textbf{v} \). In general, the intersection forms a lie when the planes are just rotations of one another, but all 3 of them aren't the same plane as in the case below.

- If there are two parameters, say \(\lambda\) and \(\mu\), then the solutions are of the form \( \textbf{x} = \textbf{a}+ \lambda \textbf{u} + \mu \textbf{v} \). This now represents a plane in \(\mathbb{R}^3\), which passes through \(A\) and is spanned by \( \textbf{u} \) and \( \textbf{v} \). However, this is only possible if the three planes at the start actually all coincided with one another.

I find that figure 1.1.2 of this article gives a good visual representation of them.On the other hand, when we have 2 equations, we're simply saying we now have only 2 planes in \( \mathbb{R}^3\). Because there's only 2 planes, our possibilities are severely narrowed

- The only way there can be no solutions is if the planes are parallel to one another.

- There can never be a unique solution. (Why?) If two planes in \( \mathbb{R}^3\) are not parallel then they must have infinite solutions.

- If there are two parameters, then once again the planes coincided.

- If there is only one parameter, again the intersection is a line. This is essentially every case when the planes aren't parallel with one another.