Calculating Cross Product in 3D and 7D

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SUMMARY

The discussion centers on the mathematical reasoning behind the existence of cross products in only 3D and 7D spaces, linked to the properties of Quaternions and Octonions. It highlights the role of anti-symmetric linear pairings and their connection to vector fields on spheres, as described in differential geometry. The conversation also emphasizes the peculiar behavior of determinants in matrices of varying dimensions, specifically 3x3 through 7x7, as a means to understand these phenomena.

PREREQUISITES
  • Understanding of Quaternions and Octonions
  • Familiarity with differential geometry concepts
  • Knowledge of anti-symmetric linear pairings
  • Basic matrix theory, including determinants
NEXT STEPS
  • Research the properties of Quaternions and their applications in 3D space
  • Study the role of Octonions in higher-dimensional mathematics
  • Explore differential geometry theorems related to vector fields on spheres
  • Analyze determinants of matrices and their implications in linear algebra
USEFUL FOR

Mathematicians, physicists, and students interested in advanced geometry, linear algebra, and the theoretical foundations of vector fields in various dimensions.

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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