Exceptions for Plane Waves Rules

Let’s consider plane waves propagating along $z$ direction, namely wave of this form: $vec E(ctpm z)$,$vec B(ct pm z)$.
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):

$$vec E=vec B times vec C$$ $$vec E cdotvec B=0$$

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,
$vec E=vec E(z,t)$, $vec B=vec B(z,t)$?