Equation of a Plane Containing Two Non-Parallel Coplanar Lines

Once two lines $\vec r=\vec a+s\vec b$ and $\vec r=\vec c+t\vec d$ are known to be coplanar (via §6.8.8), $\vec b\times\vec d\ne\vec 0$ tells us the plane's normal direction, and either base point tells us where the plane sits.

1. Parametric form. If $\vec r_0$ is the position vector of any point $P$ of that common plane, then $\vec r_0-\vec a$ (and likewise $\vec r_0-\vec c$) is coplanar with $\vec b,\vec d$, so

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

2. Non-parametric form: $(\vec r-\vec a)\cdot(\vec b\times\vec d)=0$ (equivalently, with $\vec c$ in place of $\vec a$) — the same normal $\vec b\times\vec d$ used to test coplanarity in §6.8.8 now also builds the plane's equation directly. …