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Step-by-Step Solution
Step 1: Note the Given Conditions
1. We have two non-zero complex numbers $z$ and $\omega$ such that
$|z\,\omega| = 1$. This implies
$|z|\cdot|\omega| = 1.$
Hence,
$|z| = \frac{1}{|\omega|}.$
2. The difference of their arguments is
$Arg(z) - Arg(\omega) = \frac{\pi}{2}.$
Equivalently,
$Arg\!\Bigl(\frac{z}{\omega}\Bigr) = \frac{\pi}{2}.$
Step 2: Interpret the Argument Condition
If a complex number has argument $\frac{\pi}{2}$, then it is purely imaginary (lying on the positive imaginary axis). Therefore,
$\frac{z}{\omega}$ is purely imaginary. We can write
$$\frac{z}{\omega} = k\,i \quad \text{for some real } k.$$
Step 3: Relate Magnitudes to Find $k$
Taking the modulus on both sides:
$$
\left|\frac{z}{\omega}\right| = |k\,i| = |k|\cdot|i| = |k|.
$$
But
$$
\left|\frac{z}{\omega}\right| = \frac{|z|}{|\omega|}.
$$
Since $|z| = \tfrac{1}{|\omega|},$ it follows:
$$
\frac{|z|}{|\omega|} = |z| \times \frac{1}{|\omega|} = |z|^2.
$$
Hence,
$$
|z|^2 = k \quad \text{and} \quad |z| = \sqrt{k}, \quad |\omega| = \frac{1}{\sqrt{k}}.
$$
Step 4: Find $|\overline{z}\,\omega|$
The magnitude of $\overline{z}\,\omega$ is
$$
|\overline{z}\,\omega| = |\overline{z}|\cdot|\omega| = |z|\cdot|\omega|.
$$
Since $|z|\cdot|\omega| = 1,$ we get
$$
|\overline{z}\,\omega| = 1.
$$
Step 5: Determine the Argument of $\overline{z}\,\omega$
The argument of $\overline{z}\,\omega$ is
\[
Arg(\overline{z}\,\omega) = Arg(\overline{z}) + Arg(\omega).
\]
Recall that $Arg(\overline{z}) = -\,Arg(z).$ Thus,
\[
Arg(\overline{z}\,\omega) = -\,Arg(z) + Arg(\omega)
= -\bigl(Arg(z) - Arg(\omega)\bigr)
= -\frac{\pi}{2}.
\]
An argument of $-\frac{\pi}{2}$ indicates the point lies on the negative imaginary axis.
Step 6: Conclude the Value of $\overline{z}\,\omega$
Since its magnitude is $1$ and its argument is $-\frac{\pi}{2}$, the point on the complex plane is at
$(0, -1).$
Hence,
$$
\overline{z}\,\omega = -i.
$$
Final Answer
$\boxed{-i}$
