= Solution
The phase $kz-\omega t$ describes propagation along $+z$. With $\psi=kz-\omega t$, the vacuum <plane electromagnetic wave> has
$$
\boxed{\mathbf B=\frac1c\hat{\mathbf z}\times\mathbf E=\frac Ac(-3\hat{\mathbf x}+2\hat{\mathbf y})\sin\psi.}
$$
Thus $B_x=-3A\sin\psi/c$, $B_y=2A\sin\psi/c$, and $B_z=0$. The field amplitudes are $E_0=\sqrt{13}|A|$ and $B_0=\sqrt{13}|A|/c$, giving
$$
\boxed{E_0/B_0=c.}
$$
The instantaneous magnitude ratio also equals $c$ wherever the fields are nonzero; at a node both vanish together. Their <Poynting vector> points along $+z$, checking the relative component signs.
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