Home exercise 6.1.1: A plane wave in the xy-plane

$k=3\pi a_x+4\pi a_y$ and $k=\pi \sqrt{9+16}=5\pi rad/m$, so

$E\parallel a_z$ is perpendicular to $a_k$: the wave is transverse, as required.

With $a_x\times a_z=-a_y$ and $a_y\times a_z=a_x$,

of magnitude $1/(20\pi )=15.9mA/m$.

$E\times H\propto a_z\times (0.8a_x-0.6a_y)=0.8a_y+0.6a_x=a_k$, and $|E|/|H|=6\times 20\pi =120\pi ={\eta }_0$. Also $1/{\lambda }_x^2+1/{\lambda }_y^2=2.25+4=6.25=1/{\lambda }^2$, as Eq. (6.1.20) requires.