instructions

A plane can be represented as a point plus a normal vector, but you’ll often see it sometimes represented as a plane equation. So, is there any connection between them?

The geometric

For example, in the figure below, a point B and the vector BA can represent a plane.

When you represent the plane, it doesn’t matter where B is, it can move along the plane perpendicular to AB.

equation

In algebraic terms, assume the coordinates of point A (XA, ya, za) and point B (xb, yb, Zb). (The vector AB is a constant, let’s say (xab, yab, zab))

The points O(x, y, and z) on the plane all satisfy OB ∙ AB = 0, that is, the dot product of the vectors is equal to 0. Expand the dot product equation, and obtain:

(x-xb) * xab + (y-yb) * yab + (z-zb) * zab = 0 // Since the coordinates of point B '(xb, yb, zb)' are constants, so is the vector AB '(xab, yab, zab)', So it can be simplified to xab * x + yab * y + zab * z = constantCopy the code

Let’s see, the equation of the plane is Ax + By + Cz = constant, and (A, B, C) is the plane normal, or the negative direction of the normal.