Is this an Exterior product or something else? --Salix (talk): 19:19, 27 September 2008 (UTC)[reply]

Why can't cross products be defined in fifteen dimensions? The article should provide at least some explanation, even if it's obvious, because nothing's really obvious at this level. Eebster the Great (talk) 02:36, 8 December 2008 (UTC)[reply]

I've read the attached reference to Multi-dimensional vector product by Z.K.Silagadze. The quote of interest would be: "From a view point of composition algebra,the vector product is just the commutator divided by two. According to Hurwitz theorem the only composition algebras are real numbers,complex numbers,quaternions and Octonions. Quaternions produce the usual three-dimensional vector products. The seven-dimensional vector product is generated by Octonions." I'll try to put this in with a tag for improvement. 86.80.122.213 (talk) 00:01, 3 April 2009 (UTC) LCV[reply]

Nevermind, looks like it's already there. 86.80.122.213 (talk) 00:03, 3 April 2009 (UTC) LCV[reply]

Eebster asks, 'why can we not have a fifteen dimensional vector cross product?'. Well it at least in part, the answer lies with the distributive law. The only reason why the vector product is so useful is because it can be used to decsribe certain problems in three dimensions and that we can multiply the individual components out. If the operation did not obey the distributive law, it would have no practical use.

As it so happens, the seven dimensional cross product also obeys the distributive law.

As regards the deeper question of why only three and seven, I don't know. I read in a 1970's Encyclopaedia Britannica that they were in the process of deriving a theorem to prove that it could only exist in three and seven dimensions, but that the theorem was very complicated and that it was not yet completed.

All I can say is that I once tried to multiply it out in five dimensions and it failed the consistency test. If you want to set up a 15 dimensional set of operators and try it out, you are welcome. But you will need lots of paper, and lots of patience, and your eyes will be sore at the end of it. The chances are you will make at least one silly mistake which will ruin the whole purpose of the exercise. David Tombe (talk) 03:15, 29 December 2009 (UTC)[reply]