The wave is $$\bar{E} = E_{0} \ \mathrm{sin}\left(\frac{2\pi z}{\lambda} + \omega t\right) \bar{i} + E_{0} \ \mathrm{cos} \left(\frac{2 \pi z}{\lambda}+\omega t\right) \bar{j}.$$
Let's simplify with $z = 1$. Now the $xy$-axis is defined by the parametrization $$\left(\mathrm{sin}\left(\frac{2\pi }{\lambda}+\omega t \right), \mathrm{cos}\left(\frac{2\pi }{\lambda} + \omega t\right)\right),$$ where $t$ stands for time and $\lambda$ is the wavelength. This parametrization satisfies the equation $1^2=x^{2}+y^{2}$, a circle.
Now, let's variate the value of $z$. We know now that it cannot move into $x$ or $y$ coordinates, or do we? Not really, the latter simplification is naive — $x-y$ parametrization depends on the dimension $z$ — but can we see something from it? If so, how to proceed now?
The solution is that the wave moves along the $z$-axis to the negative direction as $t$ increases, a thing I cannot see.
The way I am trying to solve this kind of problems is:
- Parametrize the equation
- suppose other things constant and change one dimension, observe
- check other variable
However, I find it hard to parametrize the $z$, so I'm a bit lost. How can I visualize the wave with pen-and-paper?