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?
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.