# How to determine wave vector given the polarization of a plane wave

This is very similar to the question asked in the past, but I need further clarification.

Say I'm given the polarization of an electromagnetic wave in free space (in phasor form), something like this:

$$\vec{E}=[2\hat{e}_{x}+3\hat{e}_{x}+(5-j)\hat{e}_{z}] e^{-j\vec{k}\cdot \vec{r}}$$

I don't particularly care about these specific coefficients (I just made them up) but lets say that $$E_x$$,$$E_y$$,$$E_z$$ are all nonzero, meaning the polarization of the $$\vec{E}$$ field isn't just in the $$xy,xz,$$ or $$yz$$ planes (as in the last post I referenced). Given this information, and knowing that $$\vec{k}$$ points in some arbitrary direction, how can I determine the direction of $$\vec{k}$$?

Here is what I have: I've been back and forth on this one for a while now, and it has been suggested to use Gauss' law in free space $$\nabla\cdot\vec{E} =0$$ which, when applied to $$\vec{E}=\vec{E}_0e^{-j\vec{k}\cdot{\vec{r}}}$$ yields:

$$-j\vec{E}\cdot\vec{k}=0$$

It is at this point I balk. Sure I can write,

$$E_xk_x+E_yk_y+E_zk_z=0$$

but even knowing this the values of $$E_x$$, $$E_y$$, and $$E_z$$ it doesn't seem like I know enough to find $$k_x$$, $$k_y$$, and $$k_z$$. What am I missing here?

I know this seems like homework, but I'm genuinely interested to know to extract $$\vec{k}$$ as it has direct application to some measurements I might make in the near future.

You cannot extract $$\vec{k}$$ from the polarization, just that its direction must be perpendicular to the polarization. There is no one-to-one matching between polarization and $$\vec{k}$$.