Solving EM Wave Questions: Polarization, Direction, Phase Velocity, Amplitude

[AwesomeTrains]

Homework Statement

Hey, I've been given this EM-wave:
$(-2\vec{e}_x+2\sqrt{3}\vec{e}_y+3\vec{e}_z)E_0e^{i[\omega t-a(\sqrt{3}x+y)]}$ with $a∈ℝ$

1) Describe the wave and how it's polarized.
2) In what direction does the wave propagate?
3) What is the phase velocity of the wave?
4) What is the amplitude?
5) Is it a transverse wave?

2. Homework Equations
1: Plane wave equation:
$E_0cos(wt-\vec{k}\vec{r}+\phi_0)$ and $e^{ix}=cos(x)+isin(x)$
3: Phase velocity:
$v_{ph}=\frac{\omega}{k}$

The Attempt at a Solution

1) I think it's a plane wave. With Euler's formula I get
$(-2\vec{e}_x+2\sqrt{3}\vec{e}_y+3\vec{e}_z)E_0[cos(\omega t-a(\sqrt{3}x+y))+isin(\omega t-a(\sqrt{3}x+y))]$
I guess $(-2\vec{e}_x+2\sqrt{3}\vec{e}_y+3\vec{e}_z)E_0cos(\omega t-a(\sqrt{3}x+y))$ is what I have to look at?

I'm really not sure about the polarization of the wave. My best guess is that it's linearly polarized, because all 3 directions have the same phase at all times.

2) I would say it moves in the direction of $\vec{k}=a(\sqrt{3},1,0)$ in the x-y plane? (I can assume that the wave propagates in vacuum when it's not stated?)

3) $v_{ph}=\frac{\omega}{|\vec{k}|}=\frac{\omega}{2|a|}$ Can I calculate the frequency ($\omega$) by what is given? Or is it $c$ since it's in vacuum. Is it in vacuum?
($v_{ph}=\frac{\omega}{k}=\frac{\lambda2\pi ν}{2\pi}=c)$
4) Pretty sure about this one:
$5E_0$

5) I've read that transverse waves are waves that are oscillating perpendicular to the direction of propagation.
And I've heard that this is a transverse wave. But $(-2,2\sqrt{3},3)$ is perpendicular to $\vec{k}=a(\sqrt{3},1,0)$ only when $a=1$ or $0$ and not for all a?

Sorry for all the questions in the post I really appreciate any answers or corrections :)

 

[DelcrossA]

You seems sound for the most part. Given the form of the wave (no absorption or anything like that), it seems to be in a vacuum with a dispersion relation ω = c k.
For question 5, think about the dot product.

 

[AwesomeTrains]

You seems sound for the most part. Given the form of the wave (no absorption or anything like that), it seems to be in a vacuum with a dispersion relation ω = c k.
For question 5, think about the dot product.

Thanks for the reply :)
The dot product between $\vec{k}=a(\sqrt3,1,0)$ and $(-2,2\sqrt3,3)$ is $a(-2)\sqrt3+2a\sqrt3+0=0 => \vec{k} \perp (-2,2\sqrt3,3)$ oh yea I see my mistake now, it works for all a. Thanks :)
Beside that is the rest correct?