The wave nature of light Light is an electromagnetic wave composed of electric and magnetic components.
I recently read that the velocity of light is in the direction $\mathbf{E}\times\mathbf{B}$ where $\mathbf{E}$ and $\mathbf{B}$ are the electric and magnetic field vectors.
Now, suppose at some point in space, the electric and magnetic fields are $\mathbf{E}$ and $\mathbf{B}$, which are both functions of time, suppose that we place a mirror there, perpendicular to the incident light, then, due to reflection from denser medium, there will be a phase difference of $\pi$ in the electric and magnetic field components of the wave.
So, the electric and magnetic fields are $-\mathbf{E}, -\mathbf{B}$ after reflection. Hence, the direction of propagation of light after reflection is the same as the direction of  $(\mathbf{-E})\times(\mathbf{-B})=\mathbf{E}\times\mathbf{B}$ which means that the light wave is still traveling in the same direction.
Where is the error in this?
 A: From Wikipedia, Reflection Phase Change

"Phase" here is the phase of the electric field oscillations, not the
magnetic field oscillations (while the electric field will undergo
180° phase change, the magnetic field will undergo 0° phase change.

A: Suppose interface to be $xy$-plane then
$$\mathbf{E}_I(z,t)=E_{0I} \ e^{i(k_1z-\omega t)}\hat{x}$$
$$\mathbf{B}_I(z,t)=\frac{1}{v_1}E_{0I} \ e^{i(k_1z-\omega t)}\hat{y}$$
It gives rise to a reflected wave,
$$\mathbf{E}_R(z,t)=E_{0R} \ e^{i(-k_1z-\omega t)}\hat{x}$$
$$\mathbf{B}_R(z,t)=-\frac{1}{v_1}E_{0R} \ e^{i(-k_1z-\omega t)}\hat{y}$$
Now if $v_2<v_1$ then there will be phase difference introducing this to reflected waves
$$\mathbf{E}_R(z,t)=E_{0R} \ e^{i(-k_1z-\omega t+\pi)}\hat{x}$$
$$\mathbf{B}_R(z,t)=-\frac{1}{v_1}E_{0R} \ e^{i(-k_1z-\omega t+\pi)}\hat{y}$$
$$\mathbf{E}_R(z,t)=-E_{0R} \ e^{i(-k_1z-\omega t)}\hat{x}$$
$$\mathbf{B}_R(z,t)=\frac{1}{v_1}E_{0R} \ e^{i(-k_1z-\omega t)}\hat{y}$$
The direction of wave propagation: $-\hat{x}\times\hat{j}=-\hat{k}$
It should be.

Reference : Sec. 9.3.2 Electrodynamics : DJ Griffith
