I think it is bad luck to try to define the geometric
product as
AB := A.B + A/\B (1)
Although alas some of the Cambridge tutorials seem to
give this impression.
As to the non-definitional equation
AB = A.B + A/\B (2)
I will concede that
-- equation (2) holds whenver A or B is a plain
grade=1 vector.
-- equation (2) holds trivially whenever A or B
is a scalar.
-- equation (2) therefore always holds in dimensionality
D=3 or lower.
BUT the fact that remains that in D=4 or higher, in
addition to the inner product A.B and the exterior
product A/\B, you can have nontrivial "middle products".
So be careful: AB does not always equal A.B + A/\B.
My tentative recommendation is to treat the geometric
product AB as primary and fundamental, and to define
A.B as the low-grade piece thereof, and A/\B as the
high-grade piece thereof. This is 100% logical and
extensible to all dimensionalities D. I see absolutely
no downside to using this approach in introductory
presentations.
(There are situations, notably in general
relativity, where you might want to define the wedge
product from scratch, when you can use neither the dot
product nor [therefore] the full geometric product.
Still, we can confidently say that
A/\B = fully antisymmetrized product (3)
whenever the RHS exists, so [unlike equation (2)]
the recommended approach can never lead to erroneous
calculations.)
This posting is the position of the writer, not that of Vlad, Drusilla,
or Spike.
This posting is the position of the writer, not that of SUNY-BSC, NAU or the AAPT.