# Cross Product

by theperthvan
Tags: cross, product
 P: 185 Why is it possible to take the cross product in only 3 and 7 dimensions?
 PF Patron Sci Advisor Emeritus P: 16,094 Er, could you clarify the question? Although I don't know precisely what you're asking, I suspect the answer has to do with the Quaternions and Octonions.
 HW Helper Sci Advisor P: 9,395 If you mean 'why do their exist anti-symmetric linear pairings x/\y : R^nxR^n-->R^n for some n, and not others', then Hurkyl is getting there. There is a theorem in differential geometry that explains this, though I don't know what it is saying (i.e. I can't encapsulate it into a nice bite sized slogan for the lay person).
HW Helper
P: 9,372

## Cross Product

existence of pairings produces vector fields on the sphere, and these exist only in a few cases. maybe this is related.
 P: n/a Well, the idea is that a matrix is created. The determinent can do very funny things. Just try to find the determinents of 3x3, 4x4, 5x5, 6x6, 7x7. You may figure out why...
 P: 185 OK. Thanks
HW Helper