Suppose that in a particular inertial frame the electric field $\EE$ and magnetic field $\BB$ are neither perpendicular nor parallel to each other.

1. Is there another inertial frame in which the fields $\EE{}”$ and $\BB{}”$ are parallel to each other?
2. Is there another inertial frame in which the fields $\EE{}”$ and $\BB{}”$ are perpendicular to each other? (You may assume without loss of generality that the inertial frames are in relative motion parallel to the $x$-axis, and that neither $\BB$ nor $\EE$ has an $x$-component. Why? Briefly justify your assumptions!)

1. Yes! Setting the cross product of (25) and (24) of §11.2 equal to zero, using the vector identity $\uu \times (\vv \times \ww) = (\uu\cdot\ww) \vv - (\uu\cdot\vv) \ww$ and the fact that $\vv$ is perpendicular to both $\EE$ and $\BB$, leads to \begin{equation} \frac{\overc{\vvs}}{1+\overcc{|\vvs|^2}} = \frac{\EE\times \cc\BB}{|\EE|^2+\csq|\BB|^2} \end{equation}

2. No; $\EE\cdot\BB$ is invariant according to (22) of §11.6.