Equation of a plane passing through 3 points
Let's assume we're given 3 distinct points in the 3-dimensional space and these points are not collinear. On a side note, this implies the given 3 points are also distinct.
Let's also assume that their coordinates in some 3-dimensional coordinate system $S = \{O,\overrightarrow{e_1},\overrightarrow{e_2},\overrightarrow{e_3}\}$ are as follows.
$P_1 = (x_1, y_1, z_1)$
$P_2 = (x_2, y_2, z_2)$
$P_3 = (x_3, y_3, z_3)$
The coordinate system $S$ does not need to be orthogonal.
Then the equation of the plane passing through these points is as follows.
$$\begin{vmatrix}x-x_1&y-y_1&z-z_1\\x_2-x_1&y_2-y_1&z_2-z_1\\x_3-x_1&y_3-y_1&z_3-z_1\end{vmatrix} = 0$$
Note: This is a determinant in the expression above.
