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).

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