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question on vector product

From: Rolf Fabian
Subject: question on vector product
Date: Wed, 10 Jan 2001 14:08:35 +0100


I've some general (more or less academical) question
concerning the possibility to define a cross -product in N-dim vector spaces.

Below, there's an excerpt from Encyclop. Britannica
I've found in the net which's related to the topic.

Vector Products

Unlike the concept of scalar product, the existence of a vector product having 
the properties
listed above depends critically upon the dimension n of the vector space on 
which it is defined.
Except for the trivial cases (n = 0 and n = 1), such a vector product exists 
only for n = 3 and n= 7. 
This odd fact is closely related to long-standing problems in algebra, 
originating in the work of
Hamilton and Cayley but only fully solved in the 1960s.

Now my  questions
1) What's the vector (cross)  prod in dimensions n=0,1 
2) What's magic with dimension n=7 ?
Can anybody give a formulae or describe an algorithm to calculate 
the7-dim vector (cross) product?

Furthermore a question not related to referenced article:
3) Even for (standard) 3-dim space, I've only found definitions for a vector 
(cross) product
defined in Euklidian 3d space (real vectors only).
What's the correct definition in unitary 3D-space (complex vectors) ?
If any .. are there more than one possibilities for such a definition ?


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