If you know two vectors on a surface, you can compute the vector that is perpendicular to these vectors by using the cross product. This perpendicular is called the normal, and it is critical to lighting surfaces properly, and in determining if a surface is facing the user (above section). How is it calculated?
Suppose you have two vectors V1(x1,y1,z1) and V2(x2,y2,z2) and you want to know what vector V3(x3,y3,z3) is perpendicular to them. First, realize that the two vectors determine a plane. This is the surface they are on. Also realize that a perpendicular line can leave the plane on the front side or the back side. This is an important consideration in calculating a normal to a surface since you want to be sure you are on the front side. Recall that the convention is that the vectors must be used in counter clockwise order to address the front side of the surface. In the diagram below you will see the correct order to produce the normal vector V3.
Without any further ado, here are the cross product calculations:
One more time. These cross product formulas allow the programmer to derive the normal to a surface given two vectors on that surface.