For each of the four following statements, indicate whether it is true or false, by justifying the answer. An unjustified answer is not taken into account. An absence of answer is not penalised.

Consider a cube ABCDEFGH with edge length 1 and the point I defined by $\overrightarrow { \mathrm { FI } } = \frac { 1 } { 3 } \overrightarrow { \mathrm { FB } }$.\\
One may place oneself in the orthonormal coordinate system of space $( \mathrm { A } ; \overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { AE } } )$.

\begin{enumerate}
  \item Consider the triangle HAC.
\end{enumerate}

Statement 1: The triangle HAC is a right-angled triangle.\\
2. Consider the lines (HF) and (DI).

Statement 2: The lines (HF) and (DI) are secant.\\
3. Consider a real number $\alpha$ belonging to the interval $] 0 ; \pi [$.

Consider the vector $\vec { u }$ with coordinates $\left( \begin{array} { c } \sin ( \alpha ) \\ \sin ( \pi - \alpha ) \\ \sin ( - \alpha ) \end{array} \right)$.\\
Statement 3: The vector $\vec { u }$ is a normal vector to the plane (FAC).\\
4. The cube ABCDEFGH has 8 vertices. We are interested in the number $N$ of segments that can be constructed by connecting 2 distinct vertices of the cube.\\
Statement 4: $N = \frac { 8 ^ { 2 } } { 2 }$.