The geometric product
The geometric product create an object called rotor.
Relationship with Dot Product and Exterior Product
We can split geometric product as the following:
The first term does not depend on the order of arguments
The dot product of two vectors is symmetric and is equal to the first part:
And the exterior product of two vectors is antisymmetric and is equal to the second part:
Note that the dot product contains the cosine of the angle between
Properties
The geometric product is defined so that vectors have inverses (i.e.
Multiplication Table
Geometric product of the basis vectors
Note that any unit vectors times itself is
For any pair of orthonormal basis vectors, the result is just the bivector they form together:
Geometric Operations with Geometric Product
Reflection
See: reflection
If we have a unit vector
We can replace the dot product
(where
Thus,
Rotation
See: two reflections give a rotation
It turns out that if we apply two successive reflections to a vector
Thus, to represent a rotation by twice the angle between the vectors
The above is the formula we use when we apply a rotation with a rotor
.