The complex plane is equipped with a direct orthonormal coordinate system $( \mathrm { O } , \vec { u } , \vec { v } )$.\\
We denote by i the complex number such that $\mathrm { i } ^ { 2 } = - 1$.\\
We consider the point A with affixe $z _ { \mathrm { A } } = 1$ and the point B with affixe $z _ { \mathrm { B } } = \mathrm { i }$.\\
To any point $M$ with affixe $z _ { M } = x + \mathrm { i } y$, with $x$ and $y$ two real numbers such that $y \neq 0$, we associate the point $M ^ { \prime }$ with affixe $z _ { M ^ { \prime } } = - \mathrm { i } z _ { M }$.\\
We denote by $I$ the midpoint of the segment [AM].\\
The purpose of the exercise is to show that for any point $M$ not belonging to (OA), the median $( \mathrm { O } I )$ of the triangle $\mathrm { OA} M$ is also a height of the triangle $\mathrm { OB } M ^ { \prime }$ (property 1) and that $\mathrm { B } M ^ { \prime } = 2 \mathrm { O } I$ (property 2).

\begin{enumerate}
  \item In this question and only in this question, we take $z _ { M } = 2 \mathrm { e } ^ { - \mathrm { i } \frac { \pi } { 3 } }$.\\
a. Determine the algebraic form of $z _ { M }$.\\
b. Show that $z _ { M ^ { \prime } } = - \sqrt { 3 } - \mathrm { i }$.\\
Determine the modulus and an argument of $z _ { M ^ { \prime } }$.\\
c. Place the points $\mathrm { A } , \mathrm { B } , M , M ^ { \prime }$ and $I$ in the coordinate system $( \mathrm { O } , \vec { u } , \vec { v } )$ using 2 cm as the graphical unit.\\
Draw the line $( \mathrm { O } I )$ and quickly verify properties 1 and 2 using the graph.
  \item We return to the general case by taking $z _ { M } = x + \mathrm { i } y$ with $y \neq 0$.\\
a. Determine the affixe of point $I$ as a function of $x$ and $y$.\\
b. Determine the affixe of point $M ^ { \prime }$ as a function of $x$ and $y$.\\
c. Write the coordinates of points $I$, B and $M ^ { \prime }$.\\
d. Show that the line $( \mathrm { O } I )$ is a height of the triangle $\mathrm { OB } M ^ { \prime }$.\\
e. Show that $\mathrm { B } M ^ { \prime } = 2 \mathrm { O } I$.
\end{enumerate}