question on vector product

[Rolf Fabian]

Hi

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
<www.britannica.com/bcom/eb/article/printable/9/0,5722,120659,00.html>
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 ?

Thanks
Rolf







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