Generalised Pythagorean Theorem?

Question

$|a+b|^2=|a|^2+|b|^2+2 Re(\overline ab)$

Can anyone explain this equality to me? How it is derived?

Answer 1

Note that $|z|^2 = z\overline{z}$. So

$|a+b|^2 = (a + b)\overline{(a+b)} = (a + b)(\overline{a} + \overline{b})$.

Now, apply the distributive property on the right hand side, and use the fact that $2 Re\ z = z+\overline{z}$.

As many pointed out, you'll realize that your original formula is not entirely correct.