Clifford Algebra is one of those things that has several names. The name “Clifford Algebra” is an older name, and the newer name is Geometric Algebra. It’s proponent/champion, David Hestenes, prefers the newer name because it is more than just an “algebra”. We might say it started out as Clifford Algebra and that the newer version of it is Geometric Algebra, with Hestenes getting credit for giving it some upgrades.
Clifford Algebra is an algebra generated by a vector space with a Quadratic Form.
Clifford Algebra fixes problems inherent in linear algebra:
- the arbitrary distinction between row vectors and column vectors
- the complex process of matrix multiplication
- the arbitrary arrangement of terms in a matrix
Clifford Algebra uses a coordinate-free representation. Motion is described with respect to a coordinate frame defined on the object, thus there is no need for an external coordinate system.
A multivector is sometimes called a Clifford Number. You can make a multivector by adding together bivectors, vectors and scalars.
The two bivectors, xz and zx correspond to the same plane segments but they have opposite orientations. For now, think of orientation as specifying a direction of travel around the perimeter (on a clock, clockwise and counterclockwise are opposite orientations).
The multiplication of two vectors (0,1) and (1,0) yields a plane segment, a square, containing the three points (0,0), (0,1) and (1,1). I think this is called a unit plane segment.
The above idea extends to 3D space where the multiplication of three vectors, (0,0,1), (0,1,0) and (0,0,1) yields a three dimensional unit segment, in this case, a cube.
Appendix A
Some miscellaneous notes will be added below…
Rob Sterling 