Space is referred to an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ).\\
We consider:
\begin{itemize}
  \item $\alpha$ any real number;
  \item the points $\mathrm { A } ( 1 ; 1 ; 0 ) , \mathrm { B } ( 2 ; 1 ; 0 )$ and $\mathrm { C } ( \alpha ; 3 ; \alpha )$;
  \item (d) the line with parametric representation:
\end{itemize}
$$\left\{ \begin{array} { l } 
x = 1 + t \\
y = 2 t , \quad t \in \mathbb { R } \\
z = - t
\end{array} \right.$$
For each of the following statements, specify whether it is true or false, then justify the answer given. An answer without justification will not be taken into account.\\
Statement 1: For all values of $\alpha$, the points $A , B$ and $C$ define a plane and a normal vector to this plane is $\vec { J } \left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)$.\\
Statement 2: There exists exactly one value of $\alpha$ such that the lines ( $A C$ ) and (d) are parallel.\\
Statement 3: A measure of the angle $\widehat { \mathrm { OAB } }$ is $135 ^ { \circ }$.\\
Statement 4: The orthogonal projection of point $A$ onto the line (d) is the point $\mathrm { H } ( 1 ; 2 ; 2 )$.\\
Statement 5: The sphere with center $O$ and radius 1 intersects the line $( d )$ at two distinct points.\\
Recall that the sphere with center $\Omega$ and radius $r$ is the set of points in space at distance $r$ from $\Omega$.