Vector Equation of Plane

Theorem
Let $P$ be a plane which passes through a point $C$ whose position vector relative to the origin $O$ is $\mathbf c$.

Let $\mathbf p$ be the vector perpendicular to $P$ from $O$.

Let $\mathbf r$ be the position vector of an arbitrary point on $P$.

Then $P$ can be represented by the equation:


 * $\mathbf p \cdot \paren {\mathbf r - \mathbf c} = 0$

where $\cdot$ denotes dot product.

Proof

 * Vector-equation-of-plane.png

It is seen that $\mathbf r - \mathbf c$ lies entirely within the plane $P$.

As $P$ is perpendicular to $\mathbf p$, it follows that $\mathbf r - \mathbf c$ is perpendicular to $\mathbf p$.

Hence by Dot Product of Perpendicular Vectors:


 * $\mathbf p \cdot \paren {\mathbf r - \mathbf c} = 0$