The discussion clarifies that the definition of the cross product remains consistent for complex vectors, particularly in the context of isomorphism between R2 and C. The geometric interpretation of the cross product, expressed as ||\mathbf{a}\times\mathbf{b}||=||\mathbf{a}|| \; ||\mathbf{b}|| \sin\theta, holds true, emphasizing the relationship between vector norms and the angle between them. However, confusion arose regarding the distinction between the cross product and the dot product, with participants confirming that the cross product is indeed perpendicular to the original vectors in three-dimensional space.
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I like Serena said:R2 and C are isomorphic.
In particular this means that the geometric interpretation is exactly the same in either R2 or C.
For Rn or Cn, the interpretation is generalized, but it's still the same.
It's still about the norm of the vectors and the cosine of the angle between them.
Are you not talking about the dot product? The cross product of two vectors in three dimensions not only has a length but is perpendicular to the two given vectors. In more than three dimensions that is not a single direction.I like Serena said:R2 and C are isomorphic.
In particular this means that the geometric interpretation is exactly the same in either R2 or C.
For Rn or Cn, the interpretation is generalized, but it's still the same.
It's still about the norm of the vectors and the cosine of the angle between them.