> It seems very interesting, but I have not currently the time to make a
> detailed comparison with vector/tensor algebra. Moreover I have not
I would suggest the freely available
Oersted Medal Lecture 2002
by David Hestenes
http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
the person who discovered and developed the Geometric Algebra.
In particular, see Section V of the above paper. It talks about
vectors, geometric products, the coordinate-free representation for
`vector product' and the geometric meaning of the imaginary unit i.
Section 1 gives a good motivation for the Geometric Algebra. Other
sections of the paper develop physical applications, from classical
mechanics to electrodynamics to non-relativistic and relativistic
quantum mechanics.
Computer Scientists might then like
http://www.geometricalgebra.net/
see the good and free introduction
http://www.geometricalgebra.net/downloads/ga4cs_chapter1.pdf
Incidentally, David Hestenes said in the lecture that he has applied
for an NSF grant to work on Geometric Algebra TWELVE times in a row,
and was rejected every single time.
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