How to prove that the intersection of two planes is a line (using parameteric equations)?

Question

I'm reading the explanation to why the intersection between two planes is a line in the textbook. This seems reasonable enough, but I don't understand the last part of the proof. This indicates that my understandingn of parametric equations may be flawed, which might come back to haunt me later.

According to the book we start with the equations of two planes, which we write as augmented matrices, so:

\begin{array}{rrr|r} A_1 & B_1 & C_1 & -D_1 \\ A_2 & B_2 & C_2 & -D_2 \end{array}

Through gaussian elimination (omitting some steps) this can be reduced to:

\begin{array}{rrr|r} 0 & 1 & a & b \\ 1 & 0 & f & g \end{array}

Which would mean that in the intersection of the two planes, the following would apply:

$$y+az=b$$ $$x+fz=g$$

The text then goes on to claim that

(1) z can be treated as a paramater (even though it should be a variable, I've never encountered such paramaters, what would that even mean?) and (2) that because of this, we could write that:

$$x=g-ft$$ $$y=b-at$$ $$z=t$$

It then goes on to clame (3) that because of this, the solution is a three dimensional line.

Can someone explain these three steps in a little more detail? Or maybe point me to a helpful guide somewhere?

Answer 1

As I understand it, you can't actually solve a system of 3 variables with only two equations, therefore you set one variable to be a parameter (in this case z) and solve for x and y. As you'll see, there are infinite parametric equations for each line, with each of them a different fixed t ∈ ℝ, but they all describe the same line.

Answer 2

Basically, getting the curve of intersection of two planes is equivalent to solving the two equations below:

\begin{gather*} A_{1} x+B_{1} y+C_{1} z=D_{1}\\ A_{2} x+B_{2} y+C_{2} z=D2\\ \end{gather*} Now, as I'm sure you know, to get the DEFINITE and UNIQUE solution to a n-variable linear system, you need at least n equations. Therefore, it is impossible to get a definite and unique solution to these equations. Hence we use a PARAMETER, to find the set of values of x,y and z that satisfy both these equations. All three of x, y and z will be defined by linear relationships to the parameter (henceforth called t). As the parameter changes, the values of x, y and z would change in such a way that they satisfy both equations 1 and 2 at all times.

Now, we know that x, y and z are defined by linear relationships with a parameter t. Well, there's a theorem in analytical geometry, that if x, y and z are related by linear relationships to the same parameter, they all lie on a line.

Hope this answers your question!