Dot Product
written by Gary Simmons
One of the most useful Formula's in 3D graphics programming is the Dot Product. It can be used for
Back face Culling and for calculating lighting for programmers creating their own lighting engine. The
Dot Product can be used to perform Back face Culling way back in World space (oh yes) to discard many
polygons, meaning they don’t have to be transformed into View space or Screen space saving a lot of
time and processing power.
OK, so I’ve sold the idea of the Dot Product to you, but what is it?
What is it?
Unlike the Cross Product, when you perform the Dot Product on two Vectors, the result is not a third
Vector. Instead, it is a scalar Value. In other words, just a single number. But what does this single
number tell us. It just so happens, that this value is actually the Cosine of the Angle between the two
Vectors, Scaled by the lengths of the Vectors.
This can be written like so:- Note V1 & V2 are both Vectors & |V1| & |V2| donates the Vector's length
Cos_angle=DotProduct(V1,V2) / |V1| * |V2|
In other words, to find the Cosine of the angle between V1 & V2 we must first perform the Dot Product
on them and then Divide the Result by the length of V1 multiplied by the Length of V2.
The length of a Vector is calculated as Follows:-Note V1 is a Vector
VectorLength=SQRT( (V1. x*V1. x) + (V1. y*V1. y) + (V1. z*v1. z) );
Note that Finding the Cosine of the angle, is even simpler if both V1 and V2 are Unit Vectors(They both
have length of 1). In this case we can do away with the scaling of the Vector Lengths because the result
of |V1|*|V2| will always be one. So, if we have two Unit Vectors , we can rewrite the above formula like
so:-
Finding the Cosine of the angle with two Unit Vectors:-
Cos_angle=DotProduct(V1,V2);<
Once we have Cos_angle , we can find the actual angle between the two Vectors with the help of the
Inverse Cosine (or ArcCosine) function. This function takes the Cosine of an Angle and returns the
Actual Angle in radians.
