Calculating Cross Product in 3D and 7D

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Discussion Overview

The discussion revolves around the mathematical concept of the cross product and its applicability in different dimensions, specifically focusing on why it is defined in 3D and 7D. Participants explore theoretical underpinnings and related mathematical structures.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions the dimensionality of the cross product, suggesting a connection to the Quaternions and Octonions.
  • Another participant refers to a theorem in differential geometry that may explain the existence of anti-symmetric linear pairings in certain dimensions, though they do not provide details.
  • A different viewpoint mentions that the existence of pairings leads to vector fields on the sphere, which are only present in specific cases.
  • One participant discusses the behavior of determinants in matrices of various sizes, implying that this could relate to the existence of the cross product in certain dimensions.
  • A later reply requests further elaboration on how determinants might explain the existence or non-existence of smooth vector fields on spheres.

Areas of Agreement / Disagreement

Participants express varying interpretations of the question and the underlying mathematical concepts, indicating that multiple competing views remain without a clear consensus.

Contextual Notes

Participants reference theorems and concepts from differential geometry and linear algebra, but do not provide complete definitions or explanations, leaving some assumptions and mathematical steps unresolved.

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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