What is the relationship between wedge product and cross product?

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SUMMARY

The wedge product and the cross product are both antisymmetric operations on vectors, but they yield different mathematical objects. The wedge product results in a bivector, which is a 2-index tensor, while the cross product produces a pseudovector. In three-dimensional vector spaces, the wedge product can be identified with the cross product due to the isomorphism between the bivector space and the vector space. This identification is non-canonical and relies on the dimensionality of the vector space.

PREREQUISITES
  • Understanding of vector spaces and their properties
  • Familiarity with antisymmetric operations
  • Knowledge of bivectors and tensors
  • Concept of pseudovectors in three-dimensional space
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  • Study the properties of bivectors in linear algebra
  • Explore the mathematical definition and applications of tensors
  • Learn about the geometric interpretation of the cross product
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Mathematicians, physics students, and anyone studying advanced vector calculus or linear algebra who seeks to understand the distinctions between different vector operations.

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What's the difference between a wedge product and a cross product?
 
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Although they are both antisymmetric in their arguments,
the wedge product of two vectors is a bivector (a 2-index tensor);
the cross product of two vectors is another [psuedo] vector.
 
Pretty much it's just down to how you view these things.

x/\y is always defined for all x,y in any vector space, they just live in the space /\^2(V). It so happens that in the case when dim(V)=3, then /\^2(V) is (non-canonically) isomorphic to V, so people identify them, and call the resulting thing the cross product.
 

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