For $z=x+\mathrm{i} y, w=u+\mathrm{i} v$, with $x, y, u, v \in \mathbb{R}$, the standard scalar product in the $\mathbb{R}$-vector space $\mathbb{C}=\mathbb{R} \times \mathbb{R}$ with respect to the basis $(1, \mathrm{i})$ is defined by
$$
\langle z, w\rangle:=\operatorname{Re}(z \bar{w})=x u+y v
$$
Verify by direct calculation that, for $z, w \in \mathbb{C}$
$$
\langle z, w\rangle^{2}+\langle\mathrm{i} z, w\rangle^{2}=|z|^{2}|w|^{2}
$$
and infer from this the CAUCHY-SCHWARZ inequality in $\mathbb{R}^{2}$ :
$$
|\langle z, w\rangle|^{2}=|x u+y v|^{2} \leq|z|^{2}|w|^{2}=\left(x^{2}+y^{2}\right)\left(u^{2}+v^{2}\right)
$$
In addition, show the following identities for $z, w \in \mathbb{C}$ by direct calculation:
$$
\begin{aligned}
|z+w|^{2} &=|z|^{2}+2\langle z, w\rangle+|w|^{2} & & \text { (cosine law) } \\
|z-w|^{2} &=|z|^{2}-2\langle z, w\rangle+|w|^{2}, & & \\
|z+w|^{2}+|z-w|^{2} &=2\left(|z|^{2}+|w|^{2}\right) & \text { (parallelogran }
\end{aligned}
$$
(parallelogram law).
Further, show that for each pair $(z, w) \in \mathbb{C}^{*} \times \mathbb{C}^{*}$ there is a unique real number $\omega:=\omega(z, w) \in]-\pi, \pi]$ with
$$
\cos \omega=\cos \omega(z, w)=\frac{\langle z, w\rangle}{|z||w|}
$$
I Differential Calculus in the Complex Plane $\mathbb{C}$
$$
\sin \omega=\sin \omega(z, w)=\frac{(\mathrm{i} z, w\rangle}{|z||w|}
$$
$\omega=\omega(z, w)$ is called the oriented angle between $z$ and $w$ and will often be denoted by $\angle(z, w)$.
Show: $\quad \angle(1, \mathrm{i})=\pi / 2, \angle(\mathrm{i}, 1)=-\pi / 2=-\angle(1, \mathrm{i})$.