Equation of a Plane Passing Through Three Given Non-Collinear Points

Theorem 6.17 (a) Parametric vector form. If $A,B,C$ (position vectors $\vec a,\vec b,\vec c$) are three non-collinear points, the vector equation of the plane through them, in parametric form, is

Proof idea: take $D$ on ray $AB$ with $\vec{AD}=s(\vec b-\vec a)$ and $\vec{DP}=t(\vec c-\vec a)$ (parallel to $AC$); then $\vec{AP}=\vec{AD}+\vec{DP}$ gives the formula above, for any point $P$ (position vector $\vec r$) of the plane.

(b) Non-parametric form. Since $\vec{AB}=\vec b-\vec a$ and $\vec{AC}=\vec c-\vec a$ both lie in the plane, and (by non-collinearity) are NOT parallel, $(\vec b-\vec a)\times(\vec c-\vec a)$ is perpendicular to the plane; the plane through $A$ perpendicular to this vector is

$\big(\vec r-\vec a\big)\cdot\big((\vec b-\vec a)\times(\vec c-\vec a)\big)=0,\qquad\text{i.e.}\qquad [\vec r-\vec a,\ \vec b-\vec a,\ \vec c-\vec a]=0.$ …