| Discovery or invention? I'm sure it's a little of both. However,
the mathematical meaning is surely pure invention.
In particular, when you take the dot product of two vectors,
it is most accurate to think of one of the vectors as a linear
functional operating on the other one to produce a scalar.
It just happens that the space of linear functionals is also
a vector space in its own right, so in engineering you can actually
be unaware of the difference most of the time. But whenever
you take a dot product, you are really combining a differential
form (which is like the contour lines on a map) with a true vector -
and the density of the contour lines crossed by your vector is the
scalar answer that comes out.
Also, its an interesting coincedance that the cross product is
isomorphic to a vector in 3 dimensions (this is called the Hodge
isomorphism in tensor analysis) - but this is not true if N is not
equal to 3.
Handedness is an interesting subject - does anyone remember the
'spinor spanner'?
- Jim
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| The handedness is related to your co-ordinate system. By convention
we use a right-handed co-ordinate system, so we get right handed
cross products. Use a left-handed co-ordinate system and you'll
get left-handed cross products.
My favorite part of vector products was the discovery that the "triple
product":
(A x B) . C
of 3-dimensional vectors is not only
commutative (unlike most vector products), but the scalar result
is equal to the volume of the parallepiped defined by the same 3
vectors in 3-space.
-- Barry
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