# Vector equation of a plane

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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

$ax+by+cz=ax_0+by_0+cz_0$