Hence show that |𝑍 − 𝜔|^2 = |𝑍|^2 − 2𝑅𝑒𝑍𝜔^-+ |𝜔|^2
. 

To show that \(|Z - \omega|^2 = |Z|^2 - 2\Re(Z\overline{\omega}) + |\omega|^2\), we will use the definition of the modulus and properties of complex numbers.

### Step-by-Step Proof:

1. **Definition of Modulus Squared**:
   For any complex number \(A\), the square of its modulus is given by:
   \[
   |A|^2 = A \cdot \overline{A}.
   \]
   Thus, for \(|Z - \omega|^2\), we have:
   \[
   |Z - \omega|^2 = (Z - \omega) \cdot \overline{(Z - \omega)}.
   \]

2. **Expand the Expression**:
   Let’s expand \((Z - \omega) \cdot \overline{(Z - \omega)}\):
   \[
   |Z - \omega|^2 = (Z - \omega)(\overline{Z - \omega}).
   \]
   Applying the distributive property, we get:
   \[
   |Z - \omega|^2 = (Z - \omega)(\overline{Z} - \overline{\omega}) = Z\overline{Z} - Z\overline{\omega} - \omega\overline{Z} + \omega\overline{\omega}.
   \]

3. **Use the Modulus Squared Property**:
   From the definition of modulus squared for complex numbers:
   \[
   Z\overline{Z} = |Z|^2 \quad \text{and} \quad \omega\overline{\omega} = |\omega|^2.
   \]
   Substituting these into the expression, we have:
   \[
   |Z - \omega|^2 = |Z|^2 - Z\overline{\omega} - \omega\overline{Z} + |\omega|^2.
   \]

4. **Combine Terms Using the Real Part**:
   Notice that \(Z\overline{\omega}\) and \(\omega\overline{Z}\) are conjugates of each other. The sum of a complex number and its conjugate equals twice its real part:
   \[
   Z\overline{\omega} + \overline{Z\overline{\omega}} = Z\overline{\omega} + \omega\overline{Z} = 2\Re(Z\overline{\omega}).
   \]
   Therefore, the expression simplifies to:
   \[
   |Z - \omega|^2 = |Z|^2 - 2\Re(Z\overline{\omega}) + |\omega|^2.
   \]

### Conclusion:
We have shown that:
\[
|Z - \omega|^2 = |Z|^2 - 2\Re(Z\overline{\omega}) + |\omega|^2.
\]

This identity is useful in various complex analysis and geometry problems, particularly when dealing with distances and angles in the complex plane.

### Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How can this identity be used to prove the triangle inequality in complex numbers?
2. What is the geometric interpretation of the expression \(|Z - \omega|\) in the complex plane?
3. How does this formula change if \(Z\) and \(\omega\) are purely imaginary?
4. Can you derive a similar identity involving the imaginary parts of \(Z\) and \(\omega\)?
5. How does this identity apply to vectors in the complex plane?

**Tip:** When working with complex numbers, breaking down expressions using their real and imaginary components often simplifies the analysis, especially when dealing with products and conjugates.

Learn how to prove the complex number identity |Z - ω|^2 = |Z|^2 - 2Re(Z * conjugate(ω)) + |ω|^2 with detailed steps and explanations. This identity is crucial in complex analysis and geometry, particularly in calculating distances and angles in the complex plane. Suitable for advanced high school students and undergraduate-level studies.

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