= Solution
Since $\mathbf E\cdot\mathbf e_z=0$, the <vector triple product> yields $\mathbf E\times(\mathbf e_z\times\mathbf E)=|\mathbf E|^2\mathbf e_z$. Consequently the <Poynting vector> is
$$
\mathbf S=\frac{E_0^2k}{\mu_0\omega}\cos^2(kz-\omega t)\mathbf e_z.
$$
Its average over a full period uses $\langle\cos^2\rangle=1/2$:
$$
\boxed{\langle\mathbf S\rangle=\frac{E_0^2k}{2\mu_0\omega}\mathbf e_z.}
$$
This is the mean electromagnetic energy flux per unit area. For positive frequency it points in the propagation direction and has magnitude $E_0^2/(2\mu_0c)=\epsilon_0cE_0^2/2$.
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