# Cross product of complex vectors

1. Jun 5, 2013

### 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?

2. Jun 5, 2013

### SteamKing

Staff Emeritus
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.

3. Jun 5, 2013

### mathman

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

4. Jun 5, 2013