To prove the given equation algebraically, we start with the left-hand side of the equation and simplify it using properties of complex numbers.

Problem: Show that for $v, w \in \mathbb{C}$,

$|z + w|^2 = |z|^2 + |w|^2 + 2\text{Re}(z \overline{w}).$

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Proof:

1. Start with the definition of the modulus squared: The modulus squared of a complex number is given by: $|z + w|^2 = (z + w)(\overline{z + w}).$

2. Expand using distributive property: Substitute $\overline{z + w} = \overline{z} + \overline{w}$: $|z + w|^2 = (z + w)(\overline{z} + \overline{w}).$

3. Multiply out the terms: Expand the product: $|z + w|^2 = z\overline{z} + z\overline{w} + w\overline{z} + w\overline{w}.$

4. Group the terms: Recognize that $z\overline{z} = |z|^2$ and $w\overline{w} = |w|^2$. So: $|z + w|^2 = |z|^2 + |w|^2 + (z\overline{w} + w\overline{z}).$

5. Simplify using the real part: Recall that for any complex number $a$, $a + \overline{a} = 2\text{Re}(a)$. Here, $z\overline{w} + w\overline{z}$ is the sum of a complex number and its conjugate, so: $z\overline{w} + w\overline{z} = 2\text{Re}(z\overline{w}).$

6. Combine the results: Substitute this back into the equation: $|z + w|^2 = |z|^2 + |w|^2 + 2\text{Re}(z\overline{w}).$

Thus, we have shown that: $|z + w|^2 = |z|^2 + |w|^2 + 2\text{Re}(z\overline{w}).$

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Explanation:

The key steps involve:

1. Expanding the definition of the modulus squared.
2. Using the property of the conjugate and the distributive law.
3. Simplifying the result by identifying the real part.

Let me know if you'd like further clarifications on any step!

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Follow-Up Questions:

1. What is the geometric interpretation of $|z|^2$ for a complex number $z$?
2. How does the real part $2\text{Re}(z\overline{w})$ affect the value of $|z + w|^2$?
3. What properties of conjugates were used in the proof?
4. How does this result connect to the triangle inequality in complex analysis?
5. Can this formula be extended to vectors in higher dimensions?

Tip:

Remember that the modulus $|z|$ of a complex number represents its distance from the origin in the complex plane.