The Vector Equation of Plane Passing through Three Non-collinear Points

Theorem 6.11. The plane through three non-collinear points $A(\vec a)$, $B(\vec b)$ and $C(\vec c)$ has the equation $\left(\vec r - \vec a\right)\cdot\left[\left(\vec b - \vec a\right)\times\left(\vec c - \vec a\right)\right] = 0$.

Why this is true: for a general point $P(\vec r)$ of the plane, the three chords $\overrightarrow{AP}$, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ all lie in the plane, so they are coplanar, which means the scalar triple product $\overrightarrow{AP}\cdot\overrightarrow{AB}\times\overrightarrow{AC}$ is zero. Writing $\overrightarrow{AP} = \vec r - \vec a$, $\overrightarrow{AB} = \vec b - \vec a$ and $\overrightarrow{AC} = \vec c - \vec a$ turns this into the stated equation (Fig. 6.10 shows all three points, the chords $\overrightarrow{AB}$ and $\overrightarrow{AC}$ drawn from $A$, and their cross product standing perpendicular to the plane as its normal). In effect, $\overrightarrow{AB} \times \overrightarrow{AC}$ plays the role that $\vec n$ or $\vec b \times \vec c$ played in the previous two theorems — three points automatically supply two directions lying in the plane, namely $\overrightarrow{AB}$ and $\overrightarrow{AC}$, and any two non-parallel directions in a plane fix its normal by taking their cross product.

Expressed in coordinates, with $A(x_1,y_1,z_1)$, $B(x_2,y_2,z_2)$, $C(x_3,y_3,z_3)$ and a general point $P(x,y,z)$, the same statement becomes the determinant equation

$\begin{vmatrix} x-x_1 & y-y_1 & z-z_1 \\ x_2-x_1 & y_2-y_1 & z_2-z_1 \\ x_3-x_1 & y_3-y_1 & z_3-z_1\end{vmatrix} = 0.$ …