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If $z$ and $w$ are two non-zero complex numbers such that $|z w|=1$, and $\operatorname{Arg}(z)-\operatorname{Arg}(w)=\dfrac{\pi}{2}$, then $\bar{z} \omega$ is equal to

1) 1
2) $-1$
3) $i$
4) $-i$
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(2)

Explanation

$|z w|=|z||\vec{w}|=|z||w|=1$
$\bar{z} w=|z| e^{-i \arg z} \cdot|w| e^{i \arg w}$
$=e^{-i(\arg z-\arg w)}$
$=e^{-i \frac{4}{2}}=-1$

by Diamond (75,948 points)

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