How to expand $\mid\mu_1\overline{\nu}_1 + \mu_2\overline{\nu}_2\mid^2$ where $\mu_i$ and $\nu_i$ are complex inner products for $i=1,2$

Question

How can we show this equality for a complex inner product $\langle \cdot, \cdot \rangle$?

$$\mid\mu_1\overline{\nu}_1 + \mu_2\overline{\nu}_2\mid^2 = \mid\mu_1\nu_1\mid^2 + \mid \mu_2\nu_2\mid^2 + 2\Re\{\mu_1\overline{\nu}_1 \overline{\mu}_2 \nu_2\},$$

where $\mu_1= \langle e_1,y \rangle, \mu_2= \langle e_2,y \rangle, \nu_1= \langle e_1,z \rangle,$ and $\nu_2= \langle e_2,z \rangle$.

And why do we take the real part of $\mu_1\overline{\nu}_1 \overline{\mu}_2 \nu_2$?

Answer 1

Because\begin{align}\bigl|\mu_1\overline{\nu_1}+\mu_2\overline{\nu_2}\bigr|^2&=\bigl(\mu_1\overline{\nu_1}+\mu_2\overline{\nu_2}\bigr)\overline{\bigl(\mu_1\overline{\nu_1}+\mu_2\overline{\nu_2}\bigr)}\\&=\bigl(\mu_1\overline{\nu_1}+\mu_2\overline{\nu_2}\bigr)\bigl(\overline{\mu_1}\nu_1+\overline{\mu_2}\nu_2\bigr)\\&=\bigl|\mu_1\nu_1\bigr|^2+\bigl|\mu_2\nu_2\bigr|^2+\mu_1\overline{\nu_1}\,\overline{\mu_2}\nu_2+\overline{\mu_1\overline{\nu_1}\,\overline{\mu_2}\nu_2}\\&=\bigl|\mu_1\nu_1\bigr|^2+\bigl|\mu_2\nu_2\bigr|^2+2\operatorname{Re}\left(\mu_1\overline{\nu_1}\,\overline{\mu_2}\nu_2\right).\end{align}