Topics: Geometry - Plane - Point - 3D Space

Planes and points can be related in several ways.

Plane that Passes Through the Origin

When (the origin), then the plane takes the vectorial representation:

We call this the plane that passes through the origin and is generated by the vectors and .

Does a Point Exist in a Plane?

To see if is in a given plane , then we can try to solve the equation system:

(remember we can write )

If the system can be solved, then the point exists in the plane. If it can’t be solved, then it doesn’t exist in the plane.

Plane that Passes Through 3 Points

Given 3 points , , , we can find the plane that passes through these three points by doing the following:

  1. Choosing one point as the base point. Let’s say we choose
  2. Calculating the direction vectors by subtracting the base point from the other points:
  3. We can use the base point and these two vectors to represent the plane that passes through these three points