Equation of a Plane Through a Point Parallel to Two Given Vectors

Let the plane pass through a given point $A$ (position vector $\vec a$) and be parallel to two given non-parallel vectors $\vec b$ and $\vec c$. If $P$ (position vector $\vec r$) is any point of the plane, then $\vec r-\vec a$ and $\vec c$ are both parallel to the plane, so $\vec r-\vec a$ lies in the plane spanned by $\vec b,\vec c$ — meaning there exist scalars $s,t$ with $\vec r-\vec a=s\vec b+t\vec c$.

1. Parametric form: $\vec r=\vec a+s\vec b+t\vec c,\quad s,t\in\mathbb R.$
2. Non-parametric form. Equivalently, $(\vec r-\vec a)\cdot(\vec b\times\vec c)=0$ (since $\vec r-\vec a$, being a combination of $\vec b,\vec c$, is automatically perpendicular to their cross product). …