The geometric product (denoted without a symbol between) is defined as the sum of a scalar from dot product and a bivector from exterior 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 and anymore (the “symmetric” part), which the second term is called the “antisymmetric” part.

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 and while the bivector contains sine of the angle, so together they fully describe the angle between the vectors as well as the plane they form. 1

Properties

The geometric product is defined so that vectors have inverses (i.e. ) and have nice properties like associativity (). 1

Multiplication Table

Geometric product of the basis vectors has the following multiplication table:

1

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 and a vector , we can reflect the vector by a plane perpendicular to as following:

We can replace the dot product with the geometric product version : 1

(where since is a unit vector)

Thus, represents reflecting the vector around .

Rotation

See: two reflections give a rotation

It turns out that if we apply two successive reflections to a vector (using vector and vector ), we get **a rotation by twice the angle between the vectors and $\mathbf{b}$$:

two reflection gives a rotation.png

Thus, to represent a rotation by twice the angle between the vectors and , we can perform the geometric products as following:

The above is the formula we use when we apply a rotation with a rotor .

Footnotes

  1. Let’s remove Quaternions from every 3D Engine (An Interactive Introduction to Rotors from Geometric Algebra) - Marc ten Bosch 2 3 4