Vector equation of a plane

In three-dimensional Euclidean space, a plane may be characterized by a point contained in the plane and a vector that is perpendicular, or normal, to the plane.

The equation of the plane containing the point $(x_{0},y_{0},z_{0})$ and perpendicular to the vector $\langle a,b,c\rangle$ is

$\langle a,b,c\rangle \cdot [\langle x,y,z\rangle -\langle x_{0},y_{0},z_{0}\rangle ]=0$

(The dot represents the dot product.)

Using the notation ${\vec {r}}=\langle x,y,z\rangle$ , ${\vec {r}}_{0}=\langle x_{0},y_{0},z_{0}\rangle$ , and ${\vec {n}}=\langle a,b,c\rangle$ , the expression becomes

${\vec {n}}\cdot ({\vec {r}}-{\vec {r}}_{0})=0$

or

${\vec {n}}\cdot {\vec {r}}={\vec {n}}\cdot {\vec {r}}_{0}$ .

(The vector $n$ is typically called the normal vector.)

Expanding and simplifying the expression, one obtains:

$a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0$

or

$ax+by+cz=ax_{0}+by_{0}+cz_{0}$