There are five ways that three distinct planes can relate to one another:
|I||none||all normals collinear|
|II||2 lines||two normals collinear|
|III||3 lines||all normals coplanar|
|IV||1 line||all normals coplanar|
|V||1 point||normals not coplanar|
Strictly speaking, systems of type II and III really have no intersection because the three planes do not share any points at all. It might seem inconsistent to talk about these lines of intersection between two of the three planes, since we ignore them in type V (which has many such lines in addition to the point), but it is just for the purpose of identifying the type.
When we say two normals `vec n_1` and `vec n_2` are collinear, we mean `vec n_1 = kvec n_2`. If all three are collinear, as in type I, then we must have `vec n_1 = kvec n_2 = cvec n_3`, where k and c are constants. Recall that three vectors are coplanar if and only if `vec n_1 * vec n_2 xx vec n_3 = 0`, and the order of the vectors does not matter.
Here is how we should go about determining the type of a system:
In the case of type IV, you can find the equation of the line of intersection by abandoning the matrix once you get a row zeroed out and rewrite the other two rows as equations. One variable will be in both equations, and the other two will not. Suppose this variable is x. Write “let `x=t`,” then replace x with t in the equations and solve for y and z. You now have the parametric equations of a line. Its direction should be collinear with the result of crossing any two of the normal vectors.