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The magnitude of a complex number, often denoted as \(|z|\), is an essential concept. It represents the distance from the origin to the point in the complex plane where the number is located. If you have a complex number written in the form \(z = a + bi\), where \(a\) and \(b\) are real numbers, its magnitude can be calculated with the formula \(|z| = \sqrt{a^2 + b^2}\). This resembles the Pythagorean theorem, which helps us understand why the magnitude is sometimes called the "length" of the complex number.
Understanding magnitudes becomes crucial when dealing with multiplication of complex numbers. Given two complex numbers \(z\) and \(\omega\), the property \(|z\omega| = |z| \cdot |\omega|\) holds. This implies that the size, or "stretch," of the complex product is the product of the individual magnitudes.
In the exercise, we are provided that \(|z\omega| = 1\). Consequently, it implies \(|z| \cdot |\omega| = 1\), leading to the relation \(|z| = \frac{1}{|\omega|}\). This tells us one number is the reciprocal of the other's magnitude, thus balancing each other out to achieve the unit magnitude of 1.

The argument of a complex number is the angle formed between the positive real axis and the line representing the complex number in the complex plane. For a complex number \(z = a + bi\), the argument \(\arg(z)\) is calculated using trigonometry, usually with \(\tan^{-1}(\frac{b}{a})\). This angle is a measure of the direction of the complex number.
When working with properties of arguments, it's critical to note that they transform under division as \(\text{arg}(z/\omega) = \text{arg}(z) - \text{arg}(\omega)\). In the given exercise, we stated that \(\arg(z) - \arg(\omega) = \frac{\pi}{2}\). This tells us that if we were to divide one complex number by another, the resulting complex number's direction is rotated by \(90^\circ\).
Having this understanding allows us to interpret complex numbers geometrically as rotations and scalings on the complex plane, aiding in solving algebraic complex problems in a more intuitive way.

Expressing complex numbers in polar form is a valuable approach for simplifying multiplication and division. A complex number \(z = a + bi\) can also be written as \(z = r e^{i\theta}\), where \(r = |z|\) is the magnitude and \(\theta = \arg(z)\) is the argument.
In polar form, multiplication of complex numbers becomes straightforward because it involves multiplying magnitudes and adding angles. Specifically, if \(z = r_z e^{i\theta_z}\) and \(\omega = r_\omega e^{i\theta_\omega}\), then their product is given by \(z\omega = (r_z \cdot r_\omega) e^{i(\theta_z + \theta_\omega)}\).
For the exercise, using the polar form, we express that when dividing the complex numbers \(z/\omega = e^{i(\frac{\pi}{2})}\) results in \(i\). This provides clarity that \(z = i\omega\), allowing the angle between \(z\) and \(\omega\) to accurately reflect a \(90^\circ\) rotation.

Multiplying complex numbers can initially seem daunting, but it becomes more intuitive by using their polar representation. As complex numbers are products of both size and direction, multiplication combines these by multiplying their magnitudes and adding their arguments.
Remember, for basic rectangular form \(z = a + bi, \omega = c + di\), multiplication follows the distributive property: \(z\omega = (ac-bd) + (ad+bc)i\). However, it's often more elegant to multiply in polar form as mentioned, which streamlines your work by transforming algebraic manipulation into simple multiplications and sum of angles.
The provided exercise showcases the product \(z\omega\), underlined by the equation \([-i(|\omega|^2)\). The identity \(\bar{z} \omega = -i\) involves complex conjugates, ensuring a deeper understanding that multiplying by conjugates contracts the angle to the opposite quadrant of the circle, thus permitting a negative result in the imaginary axis as indicated by option (c). This aligns with the property \(|z\omega| = 1\), confirming the correctness of given solutions through this geometric perspective.