Space is equipped with an orthonormal coordinate system $( O ; \vec { i } ; \vec { j } ; \vec { k } )$. We consider two lines $d _ { 1 }$ and $d _ { 2 }$ defined by the parametric representations:

$$d _ { 1 } : \left\{ \begin{array} { l } 
{ x = 2 + t } \\
{ y = 3 - t } \\
{ z = t }
\end{array} , t \in \mathbb { R } \text { and } \left\{ \begin{array} { l } 
x = - 5 + 2 t ^ { \prime } \\
y = - 1 + t ^ { \prime } \\
z = 5
\end{array} , t ^ { \prime } \in \mathbb { R } . \right. \right.$$

We admit that the lines $d _ { 1 }$ and $d _ { 2 }$ are non-coplanar. The purpose of this exercise is to determine, if it exists, a third line $\Delta$ that is simultaneously secant to both lines $d _ { 1 }$ and $d _ { 2 }$ and orthogonal to these two lines.

\begin{enumerate}
  \item Verify that the point $\mathrm { A } ( 2 ; 3 ; 0 )$ belongs to the line $d _ { 1 }$.
  \item Give a direction vector $\overrightarrow { u _ { 1 } }$ of the line $d _ { 1 }$ and a direction vector $\overrightarrow { u _ { 2 } }$ of the line $d _ { 2 }$. Are the lines $d _ { 1 }$ and $d _ { 2 }$ parallel?
  \item Verify that the vector $\vec { v } ( 1 ; - 2 ; - 3 )$ is orthogonal to the vectors $\overrightarrow { u _ { 1 } }$ and $\overrightarrow { u _ { 2 } }$.
  \item Let $P$ be the plane passing through point A, and directed by the vectors $\overrightarrow { u _ { 1 } }$ and $\vec { v }$. In this question we study the intersection of the line $d _ { 2 }$ and the plane $P$.\\
  a. Show that a Cartesian equation of the plane $P$ is: $5 x + 4 y - z - 22 = 0$.\\
  b. Show that the line $d _ { 2 }$ intersects the plane $P$ at the point $\mathrm { B } ( 3 ; 3 ; 5 )$.
  \item We now consider the line $\Delta$ directed by the vector $\vec { v} \left( \begin{array} { c } 1 \\ - 2 \\ - 3 \end{array} \right)$, and passing through the point $\mathrm { B } ( 3 ; 3 ; 5 )$.\\
  a. Give a parametric representation of this line $\Delta$.\\
  b. Are the lines $d _ { 1 }$ and $\Delta$ secant? Justify your answer.\\
  c. Explain why the line $\Delta$ answers the problem posed.
\end{enumerate}