Plane

Definition

Equation for a Plane

The planes through $P_0(x_0,y_0,z_0)$ normal to $\mathbf{n}=A\mathbf{i}+B\mathbf{j}+C\mathbf{k}$

Vector Equation

$\mathbf{n}\cdot (P_0-P)=0⟺\mathbf{nP}=\mathbf{n}\cdot {\mathbf{P}}_0=\mathbf{d}$

Component Equation

$A(x-x_0)+B(y-y_0)+C(z-z_0)=0$ or $Ax+By+Cz=D$ where $D=A{x}_0+B{y}_0+C{y}_0$

Facts

Distance from a Point to a Plane

$d=\left|PS\cdot \frac{\mathbf{n}}{n}\right|$ where $P$ is a point on a plane with normal $\mathbf{n}$ and $S$ is any point.