Definite: Plane Equation

Define:Plane Equation

...

Transformation of Format

a vector equation for plane-P 

Definition

--by Dan Sloughter 2001

Another Source Tutorial

By songho.ca

Plane Equation

..Distance from Origin

..Distance from any a Point

vector equation of the plane

let's start by assuming that we know a point that is on the plane [point]P0 .
let's also suppose that we have a vector that is orthogonal/perpendicular to the plane [vector]n .This vector is called normal vector. Now assume that [point]P is any point in the plane. Finally, since we are going to be working with vectors initially we'll let [position]r0 and [position]r for [point]P0 and [position]P respectively.
here is a sketch of all these vectors.

Notice that we added in the [vector](r-r0) which will lie completely in the plane.
Also notice that we put the normal vector on the plane, but there is actually no reason to expect this to be the case. We put it here to illustrate the point. It is completely possible that the normal vector does not touch the plane in any way.

Now, because [vector]n is orthogonal to the plane, it's also orthogonal to any vector that lies in the plane. In particular it's orthogonal to [vector](r-r0) . Recall from the Dot-Product section that two orthogonal vectors will have a dot product of zero. In other words,

scalar equation of plane

Start with the first form of the vector equation and write down a vector for the difference.