Key Concepts and Formulas

• Polar Form of a Complex Number: $z = re^{i\theta}$, where $r = |z|$ and $\theta = \arg(z)$.
• Properties of Magnitude and Argument: $|z_1z_2| = |z_1||z_2|$, $\arg(z_1z_2) = \arg(z_1) + \arg(z_2)$, $|\overline{z}| = |z|$, $\arg(\overline{z}) = -\arg(z)$.
• Complex Conjugate: If $z = x + iy$, then $\overline{z} = x - iy$.

Step-by-Step Solution

Step 1: Determine the expression for $\overline{z}\omega$ using the given information.

We are given that $|z\omega| = 1$ and $\arg(z) - \arg(\omega) = \frac{3\pi}{2}$. We want to find an expression for $\overline{z}\omega$. Let $z = r_1e^{i\theta_1}$ and $\omega = r_2e^{i\theta_2}$, where $r_1 = |z|$, $r_2 = |\omega|$, $\theta_1 = \arg(z)$, and $\theta_2 = \arg(\omega)$. Then $\overline{z} = r_1e^{-i\theta_1}$. $\overline{z}\omega = r_1e^{-i\theta_1}r_2e^{i\theta_2} = r_1r_2e^{i(\theta_2 - \theta_1)} = |z||\omega|e^{-i(\theta_1 - \theta_2)}$ Since $|z\omega| = |z||\omega| = 1$ and $\arg(z) - \arg(\omega) = \frac{3\pi}{2}$, we have: $\overline{z}\omega = 1 \cdot e^{-i(\frac{3\pi}{2})} = e^{-i\frac{3\pi}{2}} = \cos\left(-\frac{3\pi}{2}\right) + i\sin\left(-\frac{3\pi}{2}\right) = 0 + i(1) = i$ Thus, $\overline{z}\omega = i$.

Explanation: We used the polar form of complex numbers and the properties of magnitudes and arguments to simplify the expression $\overline{z}\omega$. The given information was crucial in simplifying the expression to $i$.

Step 2: Substitute $\overline{z}\omega = i$ into the expression $\frac{1 - 2\overline{z}\omega}{1 + 3\overline{z}\omega}$.

We are asked to find the argument of $\frac{1 - 2\overline{z}\omega}{1 + 3\overline{z}\omega}$. Substituting $\overline{z}\omega = i$, we get: $\frac{1 - 2i}{1 + 3i}$

Explanation: This step replaces the complex term $\overline{z}\omega$ with its simplified value $i$, transforming the problem into simplifying a standard complex fraction.

Step 3: Simplify the complex fraction $\frac{1 - 2i}{1 + 3i}$ by multiplying the numerator and denominator by the conjugate of the denominator.

The conjugate of $1 + 3i$ is $1 - 3i$. Multiplying the numerator and denominator by $1 - 3i$, we have: $\frac{1 - 2i}{1 + 3i} = \frac{(1 - 2i)(1 - 3i)}{(1 + 3i)(1 - 3i)} = \frac{1 - 3i - 2i + 6i^2}{1 - 9i^2} = \frac{1 - 5i - 6}{1 + 9} = \frac{-5 - 5i}{10} = -\frac{1}{2} - \frac{1}{2}i$

Explanation: This is a standard procedure for dividing complex numbers. Multiplying by the conjugate of the denominator eliminates the imaginary part from the denominator, allowing us to express the complex number in the $x+iy$ form.

Step 4: Find the principal argument of the complex number $-\frac{1}{2} - \frac{1}{2}i$.

Let $w = -\frac{1}{2} - \frac{1}{2}i$. We want to find $\arg(w)$. Since both the real and imaginary parts are negative, $w$ lies in the third quadrant. The reference angle $\alpha$ is given by: $\alpha = \arctan\left(\left|\frac{-\frac{1}{2}}{-\frac{1}{2}}\right|\right) = \arctan(1) = \frac{\pi}{4}$ Since $w$ is in the third quadrant, its principal argument is: $\arg(w) = -\pi + \alpha = -\pi + \frac{\pi}{4} = -\frac{3\pi}{4}$

Explanation: Identifying the correct quadrant for the complex number is essential to finding its principal argument. The principal argument is always in the range $(-\pi, \pi]$. For numbers in the third quadrant, it's typically calculated as $-\pi + \text{reference angle}$ or $-(\pi - \text{reference angle})$.

Common Mistakes & Tips

• Principal Argument Range: Always ensure your final argument lies within $(-\pi, \pi]$. Add or subtract $2\pi$ if necessary.
• Sign Errors: Be careful with signs during complex number multiplication and division, especially with $i^2 = -1$.
• Quadrant Identification: Correctly identify the quadrant of the complex number before calculating the argument.

Summary

By utilizing the polar form of complex numbers and simplifying the given expression, we found that $\arg\left( \frac{1 - 2\overline{z}\omega}{1 + 3\overline{z}\omega} \right) = -\frac{3\pi}{4}$. This corresponds to option (B).

The final answer is \boxed{{ - {{3\pi } \over 4}}}.