Consider two inertial frames \(\mathcal{O}\) and \(\mathcal{O}^{\prime}\) where \(\mathcal{O}^{\prime}\) is moving with velocity \(\mathbf{v}\) relative to \(\mathcal{O}\). We split all three-vectors in components parallel and perpendicular to the direction of the Lorentz boost, \(\mathbf{v}\) : for example, we have \(\mathbf{E}=\mathbf{E}_{\|}+\mathbf{E}_{\perp}\). Show that the Lorentz transformations of the electric and magnetic fields given in the text, equations 8.35 and 8.36 can be written instead as

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E|| E B = B B = (B (v/c E)) E =(E+((v/c) x B))