Geometric Algebra
In geometric algebra the basic elements are vectors, and these are used to build a hierarchy of elements using the definitions of products in this algebra. In particular, the algebra builds pseudovectors from vectors.
The basic multiplication in the geometric algebra is the geometric product, denoted by simply juxtaposing two vectors as in ab. This product is expressed as:
where the leading term is the customary vector dot product and the second term is called the wedge product. Using the postulates of the algebra, all combinations of dot and wedge products can be evaluated. A terminology to describe the various combinations is provided. For example, a multivector is a summation of k-fold wedge products of various k-values. A k-fold wedge product also is referred to as a k-blade.
In the present context the pseudovector is one of these combinations. This term is attached to a different mulitvector depending upon the dimensions of the space (that is, the number of linearly independent vectors in the space). In three dimensions, the most general 2-blade or bivector can be expressed as a single wedge product and is a pseudovector. In four dimensions, however, the pseudovectors are trivectors. In general, it is a (n - 1)-blade, where n is the dimension of the space and algebra. An n-dimensional space has n vectors and also n pseudovectors. Each pseudovector is formed from the outer (wedge) product of all but one of the n vectors. For instance, in four dimensions where the vectors are: {e1, e2, e3, e4}, the pseudovectors can be written as: {e234, e134, e124, e123}.
Read more about this topic: Pseudovector
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