Key Concepts and Formulas

• Modulus Properties:
  • $|z_1 z_2| = |z_1||z_2|$
  • $|\overline{z}| = |z|$
• Argument Properties:
  • $Arg(z_1 z_2) = Arg(z_1) + Arg(z_2)$
  • $Arg(\overline{z}) = -Arg(z)$
• Polar Form of a Complex Number: $z = |z|(\cos\theta + i\sin\theta)$, where $|z|$ is its modulus and $\theta = Arg(z)$ is its argument.

Step-by-Step Solution

We are given:

1. $|z\omega| = 1$
2. $Arg(z) - Arg(\omega) = \frac{\pi}{2}$ We want to find the value of $\overline{z}\omega$.

Step 1: Determine the Modulus of $\overline{z}\omega$

• We want to find the modulus of the expression. We will use the modulus properties to simplify it.
• Using the property $|z_1 z_2| = |z_1||z_2|$, we have $|\overline{z}\omega| = |\overline{z}||\omega|$.
• Using the property $|\overline{z}| = |z|$, we have $|\overline{z}\omega| = |z||\omega|$.
• Since $|z\omega| = 1$ and $|z\omega| = |z||\omega|$, we have $|z||\omega| = 1$.
• Therefore, $|\overline{z}\omega| = 1$.
• Explanation: This step shows that the magnitude of the complex number is 1.

Step 2: Determine the Argument of $\overline{z}\omega$

• We want to find the argument of the expression. We will use the argument properties to simplify it.
• Using the property $Arg(z_1 z_2) = Arg(z_1) + Arg(z_2)$, we have $Arg(\overline{z}\omega) = Arg(\overline{z}) + Arg(\omega)$.
• Using the property $Arg(\overline{z}) = -Arg(z)$, we have $Arg(\overline{z}\omega) = -Arg(z) + Arg(\omega)$.
• Factoring out -1, we have $Arg(\overline{z}\omega) = -(Arg(z) - Arg(\omega))$.
• Since $Arg(z) - Arg(\omega) = \frac{\pi}{2}$, we have $Arg(\overline{z}\omega) = -\frac{\pi}{2}$.
• Explanation: This step demonstrates that the argument of the complex number is $-\frac{\pi}{2}$.

Step 3: Combine Modulus and Argument to Find $\overline{z}\omega$

• We have the modulus $|\overline{z}\omega| = 1$ and argument $Arg(\overline{z}\omega) = -\frac{\pi}{2}$.
• Using the polar form $z = |z|(\cos\theta + i\sin\theta)$, we have $\overline{z}\omega = 1\left(\cos\left(-\frac{\pi}{2}\right) + i\sin\left(-\frac{\pi}{2}\right)\right)$.
• Since $\cos\left(-\frac{\pi}{2}\right) = 0$ and $\sin\left(-\frac{\pi}{2}\right) = -1$, we have $\overline{z}\omega = 1(0 + i(-1)) = -i$.
• Explanation: Combining the modulus and argument, we find the complex number is equal to $-i$.

Common Mistakes & Tips

• Remember the sign change when taking the argument of a conjugate: $Arg(\overline{z}) = -Arg(z)$.
• Be careful with argument arithmetic. Since arguments are angles, they are only defined up to multiples of $2\pi$.
• It's helpful to visualize complex numbers on the complex plane to understand their arguments.

Summary

By utilizing the modulus and argument properties of complex numbers and the given conditions, we determined that $|\overline{z}\omega| = 1$ and $Arg(\overline{z}\omega) = -\frac{\pi}{2}$. Converting this to rectangular form gives us $\overline{z}\omega = -i$.

Final Answer

The final answer is \boxed{-i}, which corresponds to option (A).