I am having trouble understanding how $10jy$ is converted to $10 e^{j\pi/2}$. Here $x$ and $y$ are unit vectors:
$\large=\operatorname{Re}\left[(10\hat{x}-10j\hat{y})e^{-j10\pi z}e^{jwt}\right]$ $\large=\operatorname{Re}\left[(10\hat{x}-10e^{j\pi/2}\hat{y})e^{-j10\pi z}e^{jwt}\right]$ $\large=\underbrace{10\hat{x}\cdot\cos(\omega t-10\pi z)}_{Ex}+\underbrace{10\hat{y}\cos(\omega t-10\pi z-\tfrac{\pi}{2})}_{Ey}$
Thank you.