Let ABCDEFGH be a cube. The space is referred to the orthonormal coordinate system ( A ; $\overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { AE } }$ ).

For every real $t$, consider the point $M$ with coordinates ( $1 - t ; t ; t$ ).

It is admitted that the lines (BH) and (FC) have respectively the following parametric representations:
$$\left\{ \begin{array} { l } 
{ x = 1 - t } \\
{ y = t } \\
{ z = t }
\end{array} \quad \text { where } t \in \mathbb { R } \quad \text { and } \left\{ \begin{array} { r l } 
x & = 1 \\
y & = t ^ { \prime } \\
z & = 1 - t ^ { \prime }
\end{array} \quad \text { where } t ^ { \prime } \in \mathbb { R } . \right. \right.$$

\begin{enumerate}
  \item Show that for every real $t$, the point $M$ belongs to the line (BH).
  \item Show that the lines (BH) and (FC) are orthogonal and non-coplanar.
  \item For every real $t ^ { \prime }$, consider the point $M ^ { \prime } \left( 1 ; t ^ { \prime } ; 1 - t ^ { \prime } \right)$.\\
a. Show that for all real numbers $t$ and $t ^ { \prime } , M M ^ { \prime 2 } = 3 \left( t - \frac { 1 } { 3 } \right) ^ { 2 } + 2 \left( t ^ { \prime } - \frac { 1 } { 2 } \right) ^ { 2 } + \frac { 1 } { 6 }$.\\
b. For which values of $t$ and $t ^ { \prime }$ is the distance $M M ^ { \prime }$ minimal? Justify.\\
c. Let P be the point with coordinates $\left( \frac { 2 } { 3 } ; \frac { 1 } { 3 } ; \frac { 1 } { 3 } \right)$ and Q the point with coordinates $\left( 1 ; \frac { 1 } { 2 } ; \frac { 1 } { 2 } \right)$. Justify that the line (PQ) is perpendicular to both lines (BH) and (FC).
\end{enumerate}