Comparison of vector algebra and geometric algebra

Geometric algebra (GA) is an extension or completion of vector algebra (VA). The reader is herein assumed to be familiar with the basic concepts and operations of VA and this article will mainly concern itself with operations in  the GA of 3D space (nor is this article intended to be mathematically rigorous). In GA, vectors are not normally written boldface as the meaning is usually clear from the context.

The fundamental difference is that GA provides a new product of vectors called the "geometric product". Elements of GA are graded multivectors, scalars are grade 0, usual vectors are grade 1, bivectors are grade 2 and the highest grade (3 in the 3D case) is traditionally called the pseudoscalar and designated.

The ungeneralized 3D vector form of the geometric product is:

that is the sum of the usual dot (inner) product and the outer (exterior) product (this last is closely related to the cross product and will be explained below).

In VA, entities such as pseudovectors and pseudoscalars need to be bolted on, whereas in GA the equivalent bivector and pseudovector respectively exist naturally as subspaces of the algebra.

For example, applying vector calculus in 2 dimensions, such as to compute torque or curl, requires adding an artificial 3rd dimension and extending the vector field to be constant in that dimension, or alternately considering these to be scalars. The torque or curl is then a normal vector field in this 3rd dimension. By contrast, geometric algebra in 2 dimensions defines these as a pseudoscalar field (a bivector), without requiring a 3rd dimension. Similarly, the scalar triple product is ad hoc, and can instead be expressed uniformly using the exterior product and the geometric product.