Finding vectors that direct the discriminatory factorial axis 
Let be a set $E$ of $100$ individuals for who the quantitative variables $x_1$ and $x_2$ have been observed. This set is partitioned into $C_1$ and $C_2$ which contains $40$ and $60$ individuals respectively. Each individuals have a weight of $1/100$. We write $g_1$ and $g_2$ the gravity center of $C_1$ and $C_2$ and $V$ the total variance matrix, that is to say the matrix of variable $x_1$ and $x_2$ which have been observed for each of the $100$ individuals. We assume that :
$$g_1=\begin{pmatrix}6\\1
\end{pmatrix}, g_2=\begin{pmatrix}1\\-4
\end{pmatrix}$$ 
I'm looking for the vector that direct the only discriminatory  factorial axis of the following matrix :
$$V=
\begin{pmatrix}
5 & 0\\
0 & 2
\end{pmatrix}$$

I have the following eigenvectors and values :


*

*$\begin{pmatrix}1\\0\end{pmatrix}$ for $5$

*$\begin{pmatrix}0\\1\end{pmatrix}$ for $2$


Normalizing the vector I get :


*

*$v_1=\frac{1}{\sqrt{5}}\begin{pmatrix}1\\0\end{pmatrix}$

*$v_2=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\1\end{pmatrix}$


Here is the formula to apply :
\begin{align}
\frac{g_2-g_1}{||g_2-g_1||}&=
\frac{\begin{pmatrix}1\\-4\end{pmatrix}-\begin{pmatrix}6\\1\end{pmatrix}}{||\begin{pmatrix}1\\-4\end{pmatrix}-\begin{pmatrix}6\\1\end{pmatrix}||}\\
&=\frac{-5\begin{pmatrix}1\\1\end{pmatrix}}{||-5\begin{pmatrix}1\\1\end{pmatrix}||}\\
&=-\frac{1}{2}\begin{pmatrix}1\\1\end{pmatrix}
\end{align}
