Exercise 2 (Candidates who have not followed the specialization course)

The complex plane is referred to the direct orthonormal frame $(\mathrm{O}, \vec{u}, \vec{v})$. Let $R$ be the rotation of the plane with centre $\Omega$, with affix $\omega$ and angle of measure $\theta$. The image by $R$ of a point in the plane is therefore defined as follows:
\begin{itemize}
  \item $R(\Omega) = \Omega$
  \item for any point $M$ in the plane, distinct from $\Omega$, the image $M'$ of $M$ is defined by
$$\Omega M' = \Omega M \text{ and } (\overrightarrow{\Omega M}, \overrightarrow{\Omega M'}) = \theta \quad [2\pi].$$
\end{itemize}
We recall that, for points $A$ and $B$ with affixes $a$ and $b$ respectively,
$$AB = |b - a| \text{ and } (\vec{u}, \overrightarrow{AB}) = \arg(b - a) \quad [2\pi]$$

\begin{enumerate}
  \item Show that the affixes $z$ and $z'$ of any point $M$ in the plane and its image $M'$ by the rotation $R$ are related by the relation
$$z' - \omega = \mathrm{e}^{\mathrm{i}\theta}(z - \omega).$$
  \item We consider the points I and B with affixes $z_{\mathrm{I}} = 1 + \mathrm{i}$ and $z_{\mathrm{B}} = 2 + 2\mathrm{i}$ respectively. Let $R$ be the rotation with centre B and angle of measure $\frac{\pi}{3}$.\\
  a. Give the complex form of $R$.\\
  b. Let A be the image of I by $R$. Calculate the affix $z_{\mathrm{A}}$ of A.\\
  c. Show that O, A and B lie on the same circle with centre I. Deduce that OAB is a right-angled triangle at A. Give a measure of the angle $(\overrightarrow{\mathrm{OA}}, \overrightarrow{\mathrm{OB}})$.\\
  d. Deduce a measure of the angle $(\vec{u}, \overrightarrow{\mathrm{OA}})$.
  \item Let $T$ be the translation of vector $\overrightarrow{\mathrm{IO}}$. We set $\mathrm{A}' = T(\mathrm{A})$.\\
  a. Calculate the affix $z_{\mathrm{A}'}$ of $\mathrm{A}'$.\\
  b. What is the nature of the quadrilateral OIAA'?\\
  c. Show that $-\frac{\pi}{12}$ is an argument of $z_{\mathrm{A}'}$.
\end{enumerate}