Equation of the plane

In space the plane is no longer the environment in which we work, but a geometric object with an equation of its own. It is identified by a point $P_0$ belonging to it and by a vector $\vec n$ perpendicular to it, called the normal vector.

A plane is identified by one of its points $P_0$ and by the normal vector $\vec n$.

Property — Equation of the plane

The plane passing through $P_0(x_0,y_0,z_0)$ with normal vector $\vec n=(a,b,c)$ satisfies $(\vec S - \vec S_0)\cdot\vec n = 0$, that is $ax + by + cz + d = 0, \qquad d = -(ax_0+by_0+cz_0).$ The coefficients $a,b,c$ are precisely the components of the normal vector.

Example

With $P_0(2;0;0)$ and $\vec n=(-7;-4;-1)$: $\;d=-(-7\cdot 2)=14$, so the plane is $-7x-4y-z+14=0.$

Links

Topics: Analytic geometry in space
Concepts: Equation of the plane · Plane · Normal vector
Skills: Analytic geometry · Using formulae