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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 $
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