ABSTRACT
We decompose the geometric product AB into symmetric and antisymmetric part [1]:
AB = (1/2)(AB + BA) + (1/2)(AB − BA) (2.6)
Comparing (2.2) with (2.6) and noting the relations (2.3)–(2.5), we establish that
(AB)o + (AB)4 = (1/2)(AB + BA) = (BA)o + (BA)4 and
(AB)2 = (1/2)(AB − BA) = −(BA)2. 27
Algebra and Applications to
Stated more explicitly, we can write
A · B + A∧ B = (1/2)(AB + BA) = B · A+ B ∧ A, (2.7) and
a ∧ (b · B) + a · (b ∧ B) = (1/2)(AB − BA). (2.8) The expression (1/2)(AB − BA) is called the commutator or commutator
product of A and B. In three-dimensional space
A∧ B = 0 Equation 2.2 and Equation 2.7 become
AB = (AB)0 + (AB)2 = A · B + a ∧ (b · B) + a · (b ∧ B) (2.9) and
A · B = (1/2)(AB + BA) = B · A. (2.10) The geometric product of bivectors can be generalized to the geometric
product of multivectors of any grades, Ar Bs . For the geometric product
Ar Bs = a1a2 . . . ar Bs, (r ≤ s), the term of the lowest grade will be
Ar · Bs = ( Ar Bs)s−r . (2.11) Corollary: The geometric product Ar Bs can have a nonzero scalar part
Ar · Bs if r = s. Factorization: In geometric algebra there is a type of factorizing an r -graded multivector into an outer product of vectors.
