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Re: magnitude of parallelograms and parallelepipeds



Hi Bob-
Writing AB=AdotB + A/\B is misleading, at best. It seems to imply
that the scalar AdotB can be added to the elements of A/\B, which is
wrong. The elements of the algebra are: 1, i, the three sigmas, and i
times the 3 sigmas (which are the bivectors) (note that the bivectors have
directions, but the + and - of the directions are arbitrary). Thus the
correct statement is:
AB = (AdotB)1 + A/\B
The significance of the "1" (the identity operator) becomes transparent if
you use a matrix representation of this algebra: the scalar multiplies the
identity matrix and the wedge product is a traceless matrix, so that the
scalar is given by the trace of the sum.
All of this stems, in straightforward fashion, from the identity
for Pauli matrices:
s_{i}s_{j}=delta(ij) + epsilon_{ijk}s_{k}.
It is too unwieldy to explain the details in text, I'll try to
find time to explain on my website what I think all this amounts to.
If you're familiar with the notion of projective representations of
vectors, that's really what we're dealing with.
The bivector is exactly the pseudovector in an extremely
transparent disguise. And pseudovectors are important; don't forget that
a compass needle points along a pseudovector.
Regards,
Jack

On Sat, 31 Aug 2002, Bob Sciamanda wrote:

Thanks John

My problem is interpreting the geometric product of two bivectors.
The definition for vectors: AB =A dot B + A /\ B
is not directly applicable if A and B are bivectors (?)
What do "dot" and /\ mean between bivectors?

Bob Sciamanda (W3NLV)
Physics, Edinboro Univ of PA (em)
trebor@velocity.net
http://www.velocity.net/~trebor


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