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.