Calculating Cross Product in 3D and 7D

AI Thread Summary
The discussion centers on the mathematical reasoning behind the existence of the cross product in only 3D and 7D spaces, which relates to the properties of Quaternions and Octonions. It highlights that anti-symmetric linear pairings exist in specific dimensions due to a theorem in differential geometry, which connects to the behavior of vector fields on spheres. The conversation also emphasizes the role of determinants in understanding these properties, suggesting that examining determinants of various matrix sizes can reveal insights into the existence of smooth vector fields. The complexity of determinants in higher dimensions contributes to the unique characteristics of these mathematical constructs. Understanding these concepts is crucial for grasping why certain dimensions allow for the cross product while others do not.
theperthvan
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Why is it possible to take the cross product in only 3 and 7 dimensions?
 
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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.
 
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).
 
existence of pairings produces vector fields on the sphere, and these exist only in a few cases. maybe this is related.
 
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...
 
OK. Thanks
 
prasannapakkiam said:
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...

PLease could you elaborate on why determinants of matrices can explain the (non-) existence of smooth (no-where zero, I imagine) vector fields on S^n?
 

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