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I am having trouble understanding how $10jy$ is converted to $10 e^{j\pi/2}$. Here $x$ and $y$ are unit vectors:

(original image)

$$\large=\operatorname{Re}\left[(10\hat{x}-10j\hat{y})e^{-j10\pi z}e^{jwt}\right]$$ $$\large=\operatorname{Re}\left[(10\hat{x}-10e^{j\pi/2}\hat{y})e^{-j10\pi z}e^{jwt}\right]$$ $$\large=\underbrace{10\hat{x}\cdot\cos(\omega t-10\pi z)}_{Ex}+\underbrace{10\hat{y}\cos(\omega t-10\pi z-\tfrac{\pi}{2})}_{Ey}$$

Thank you.

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up vote 2 down vote accepted

Euler's formula says that for any $\theta$, $$e^{j\theta}=\cos(\theta)+j\sin(\theta).$$ Therefore, $$e^{j\pi/2}=\cos(\tfrac{\pi}{2})+j\sin(\tfrac{\pi}{2})=0+j\cdot 1=j$$ and thus $10j=10e^{j\pi/2}$ (or, if you want to talk about vectors, $10j\hat{y}=10e^{j\pi/2}\hat{y}$; but note that it is incorrect to say that $10j\hat{y}=10e^{j\pi/2}$, because the left side is a vector, and the right side is a scalar).

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thanks ! I though that it was Euler's formula but got lost as there are many variations... – GorillaApe Jun 18 '12 at 16:24

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