Let's consider plane waves propagating along $z$ direction, namely wave of this form: $\vec E(ct\pm z)$,$\vec B(ct \pm z)$. On my book there is a derivation that shows the following relations are true for such waves ($\vec C$ intensity and direction are those of the speed of light in vacuum):
$$\vec E=\vec B \times \vec C$$ $$\vec E \cdot\vec B=0$$
I tried to use the linearity of the Maxwell equations to see if the general solution $\vec E(ct + z)+\vec E(ct - z)$ satisfies the above relations and it seems that it's not always true, however, I may have done some error.
My question is, are the above equations valid for all the plane waves, $\vec E=\vec E(z,t)$, $\vec B=\vec B(z,t)$?
