Forming equation of a plane by solving linear equation set

Question

Given three points on the plane: $ A(x_1, y_1, z_1) $, $ B(x_2, y_2, z_2) $ and $ C(x_3, y_3, z_3) $.

I'm trying to obtain the equation of the plane in this format:
$ ax + by + cz + d = 0 $

I substituted given three points into the plane equation above to form this matrix equation below:

\begin{equation} \begin{bmatrix} x_1 & y_1 & z_1 & 1 \\ x_2 & y_2 & z_2 & 1 \\ x_3 & y_3 & z_3 & 1 \\ ? & ? & ? & ? \end{bmatrix} \begin{bmatrix} a \\ b \\ c \\ d \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ ? \end{bmatrix} \end{equation}

My aim is to find the coefficients $ a $, $ b $, $ c $ and $ d $ by solving this matrix equation. However, I can't find a fourth equation to complete the equation set. Can you please write me a fourth equation to complete the set?

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Note: My aim is not just finding the plane equation. My aim is to find the plane equation by this method, by means of solving a linear set of equations. I know the other more practical way of finding the plane equation, but I'm trying to find it this way on purpose. There is no reason, I just like trying different methods and playing with numbers occasionally out of interest. So, please consider this not while writing your answers and don't suggest me other methods.

Answer 1

If you really want to solve a system of four equations (which isn't really necessary, as Brian Scott pointed out), then the fourth one that you're missing could be almost anything. For example $a+b+c = 1$ would work. The only purpose of this fourth equation is to remove the scaling indeterminacy in $a,b,c,d$ that was explained in Brian's answer.

Answer 2

The problem is that there isn’t a unique solution. Consider, for instance, the plane $x+y+z=1$: it can just as well be described by the equation $2x+2y+2z=2$. In short, if $d\ne 0$ you can always divide $ax+by+xz+d=0$ through by $d$ to get an equivalent equation with constant term $1$:

$$\frac{a}dx+\frac{b}dy+\frac{c}dz+1=0\;.$$

Thus, you can assume from the beginning that $d=0$ or $d=1$, depending on whether the plane passes through the origin or not. That reduces the problem to a system of three equations in three unknowns.

Answer 3

To get an equation of the plane you're interested in ...

(1) Put $x$, $y$, $z$, and 1 in place of your four ? symbols

(2) Take the determinant of the resulting $4 \times 4$ matrix and set it equal to 0.

You were sooooo close :-)