Equation of a Plane Through Two Points Parallel to a Vector

Let the plane pass through two given distinct points $A,B$ (position vectors $\vec a,\vec b$) and be parallel to a non-zero vector $\vec c$ that is NOT parallel to $\vec b-\vec a$ (otherwise all three directions would collapse to a line, not fix a plane).

1. Parametric form: since the plane through $A,B$ parallel to $\vec c$ is spanned by the two directions $\vec b-\vec a$ and $\vec c$,

   $\vec r=\vec a+s(\vec b-\vec a)+t\vec c\qquad\text{or equivalently}\qquad \vec r=(1-s)\vec a+s\vec b+t\vec c,\qquad s,t\in\mathbb R.$

2. Non-parametric form: $(\vec r-\vec a)\cdot\big((\vec b-\vec a)\times\vec c\big)=0$. …