Cross product for complex vectors

[dimension10]

I was wondering, is the definition of the cross product the same for complex vectors? And if it is, then how is its geometric interpretation, that is

$$||\mathbf{a}\times\mathbf{b}||=||\mathbf{a}|| \; ||\mathbf{b}|| \sin\theta$$

Thanks.

 

[I like Serena]

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.

 

[dimension10]

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.

I don't get how it is about the cosine of the angle between them. Shouldn't it be sine?

And is calculating the cross product the same as that for complex vectors? That is,

$$\left(a\hat{ \imath}+b\hat{\jmath}+c\hat{k} \right)\times \left( P\hat{\imath}+Q\hat{\jmath}+R\hat{k} \right)=\det\begin{bmatrix} \hat{\imath} & \hat{\jmath} & \hat{k} \\ a & b & c \\ P & Q & R \end{bmatrix}$$

 

[HallsofIvy]

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]

My apologies, that was not one of my brighter responses.
I did indeed mix up the dot product with the cross product.

I'm not aware of any definition of the cross product for complex vectors.