How to genarelize the vector product in three dimensions to higher dimensions?

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SUMMARY

The discussion centers on generalizing the vector product from three dimensions to higher dimensions, emphasizing that the vector product is a unique characteristic of three-dimensional space. It highlights the use of metrics to discuss volumes and hypervolumes, leading to the concept of hypercubes orthogonal to complementary dimensions. The wedge product is identified as a crucial tool for performing these operations, particularly when an orthonormal basis is established.

PREREQUISITES
  • Understanding of vector products in three dimensions
  • Familiarity with metrics in geometry
  • Knowledge of hypercubes and their properties
  • Concept of wedge products in linear algebra
NEXT STEPS
  • Research the properties of hypercubes in higher dimensions
  • Study the application of wedge products in differential geometry
  • Explore the concept of orthonormal bases in vector spaces
  • Learn about metrics and their role in defining volumes in higher dimensions
USEFUL FOR

Mathematicians, physicists, and students studying advanced geometry or linear algebra, particularly those interested in the generalization of vector operations in higher-dimensional spaces.

wdlang
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it seems that the vector product between vectors in three dimensions is peculiar property of the three dimensional space
 
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wdlang said:
it seems that the vector product between vectors in three dimensions is peculiar property of the three dimensional space

That is true.

If one has a metric you can talk about volumes and hypervolumes and come up with an a hypercube orthogonal to any given hypercube of complementary dimension. If you have an orientation then an oriented hypercube will have an oriented complement.

This all can be done with wedge product one you have an orthonormal basis.
 

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