Cross product of complex vectors

[GabDX]

How is computed the cross product of complex vectors?

Let $\mathbf{a}$ and $\mathbf{b}$ be two vectors, each having complex components.
$$\mathbf{a} = a_x \mathbf{\hat{x}} + a_y \mathbf{\hat{y}} + a_z \mathbf{\hat{z}}$$
$$\mathbf{b} = b_x \mathbf{\hat{x}} + b_y \mathbf{\hat{y}} + b_z \mathbf{\hat{z}}$$
For example, the $x$ component of $\mathbf{a}$ could be $a_x=3+4i$.

I know that the dot product of $\mathbf{a}$ and $\mathbf{b}$ is
$$\mathbf{a} \cdot \mathbf{b} = a_x b_x^* + a_y b_y^* + a_z b_z^*$$
where $^*$ denotes the complex conjugate. Is there some similar trick that should be done with the cross product of complex vectors or is it the same as with real vectors? In other words, is the cross product given by
$$\mathbf{a}\times\mathbf{b} = (a_y b_z - a_z b_y)\mathbf{\hat{x}}
+ (a_z b_x - a_x b_z)\mathbf{\hat{y}}
+ (a_x b_y - a_y b_x)\mathbf{\hat{z}}$$
or is it something different?

 

[SteamKing]

In such a vector definition, it is not clear what the unit vectors x-cap, y-cap, and z-cap would represent.

In standard complex variables, a complex function f(z) represents a mapping from the complex x,y plane to another complex u,v plane, in my understanding. Thus f(z) = f(x+iy) = u + iv.

 

[mathman]

This question has been raised before. Google "cross product complex vectors".

 

[GabDX]

Raised but not answered!

 

[CellCoree]

The cross product of complex vectors is computed in a similar way as with real vectors. The only difference is that the complex conjugate is taken for each component in the calculation. So the cross product of two complex vectors $\mathbf{a}$ and $\mathbf{b}$ is given by:

$$\mathbf{a}\times\mathbf{b} = (a_y b_z^* - a_z b_y^*)\mathbf{\hat{x}}
+ (a_z b_x^* - a_x b_z^*)\mathbf{\hat{y}}
+ (a_x b_y^* - a_y b_x^*)\mathbf{\hat{z}}$$

This is because the cross product represents a vector perpendicular to both $\mathbf{a}$ and $\mathbf{b}$, and the complex conjugate ensures that the resulting vector is also perpendicular in the complex plane. So the cross product of complex vectors is a useful tool in understanding the geometry of complex numbers.