The objective of this exercise is to determine the distance between two non-coplanar lines. By definition, the distance between two non-coplanar lines in space, $( d _ { 1 } )$ and $( d _ { 2 } )$ is the length of the segment $[\mathrm { EF }]$, where E and F are points belonging respectively to $\left( d _ { 1 } \right)$ and to $( d _ { 2 } )$ such that the line (EF) is orthogonal to $( d _ { 1 } )$ and $( d _ { 2 } )$. The space is equipped with an orthonormal coordinate system $( \mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k } )$. Let $\left( d _ { 1 } \right)$ be the line passing through $\mathrm { A } ( 1 ; 2 ; - 1 )$ with direction vector $\overrightarrow { u _ { 1 } } \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right)$ and $\left( d _ { 2 } \right)$ the line with parametric representation: $\left\{ \begin{array} { l } x = 0 \\ y = 1 + t \\ z = 2 + t \end{array} , t \in \mathbb { R } \right.$.
Give a parametric representation of the line $\left( d _ { 1 } \right)$.
Prove that the lines $\left( d _ { 1 } \right)$ and $\left( d _ { 2 } \right)$ are non-coplanar.
Let $\mathscr { P }$ be the plane passing through A and directed by the non-collinear vectors $\overrightarrow { u _ { 1 } }$ and $\vec { w } \left( \begin{array} { c } 2 \\ - 1 \\ 1 \end{array} \right)$. Justify that a Cartesian equation of the plane $\mathscr { P }$ is: $- 2 x + y + 5 z + 5 = 0$.
a. Without seeking to calculate the coordinates of the intersection point, justify that the line $( d _ { 2 } )$ and the plane $\mathscr { P }$ are secant. b. We denote F the intersection point of the line $( d _ { 2 } )$ and the plane $\mathscr { P }$. Verify that the point F has coordinates $\left( 0 ; - \frac { 5 } { 3 } ; - \frac { 2 } { 3 } \right)$. Let $( \delta )$ be the line passing through F with direction vector $\vec { w }$. It is admitted that the lines $( \delta )$ and $( d _ { 1 } )$ are secant at a point E with coordinates $\left( - \frac { 2 } { 3 } ; - \frac { 4 } { 3 } ; - 1 \right)$.
a. Justify that the distance EF is the distance between the lines $\left( d _ { 1 } \right)$ and $\left( d _ { 2 } \right)$. b. Calculate the distance between the lines $\left( d _ { 1 } \right)$ and $\left( d _ { 2 } \right)$.
\section*{Exercise 4}
The objective of this exercise is to determine the distance between two non-coplanar lines. By definition, the distance between two non-coplanar lines in space, $( d _ { 1 } )$ and $( d _ { 2 } )$ is the length of the segment $[\mathrm { EF }]$, where E and F are points belonging respectively to $\left( d _ { 1 } \right)$ and to $( d _ { 2 } )$ such that the line (EF) is orthogonal to $( d _ { 1 } )$ and $( d _ { 2 } )$.\\
The space is equipped with an orthonormal coordinate system $( \mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k } )$.\\
Let $\left( d _ { 1 } \right)$ be the line passing through $\mathrm { A } ( 1 ; 2 ; - 1 )$ with direction vector $\overrightarrow { u _ { 1 } } \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right)$ and $\left( d _ { 2 } \right)$ the line with parametric representation: $\left\{ \begin{array} { l } x = 0 \\ y = 1 + t \\ z = 2 + t \end{array} , t \in \mathbb { R } \right.$.
\begin{enumerate}
\item Give a parametric representation of the line $\left( d _ { 1 } \right)$.
\item Prove that the lines $\left( d _ { 1 } \right)$ and $\left( d _ { 2 } \right)$ are non-coplanar.
\item Let $\mathscr { P }$ be the plane passing through A and directed by the non-collinear vectors $\overrightarrow { u _ { 1 } }$ and $\vec { w } \left( \begin{array} { c } 2 \\ - 1 \\ 1 \end{array} \right)$. Justify that a Cartesian equation of the plane $\mathscr { P }$ is: $- 2 x + y + 5 z + 5 = 0$.
\item a. Without seeking to calculate the coordinates of the intersection point, justify that the line $( d _ { 2 } )$ and the plane $\mathscr { P }$ are secant.\\
b. We denote F the intersection point of the line $( d _ { 2 } )$ and the plane $\mathscr { P }$.\\
Verify that the point F has coordinates $\left( 0 ; - \frac { 5 } { 3 } ; - \frac { 2 } { 3 } \right)$.\\
Let $( \delta )$ be the line passing through F with direction vector $\vec { w }$. It is admitted that the lines $( \delta )$ and $( d _ { 1 } )$ are secant at a point E with coordinates $\left( - \frac { 2 } { 3 } ; - \frac { 4 } { 3 } ; - 1 \right)$.
\item a. Justify that the distance EF is the distance between the lines $\left( d _ { 1 } \right)$ and $\left( d _ { 2 } \right)$.\\
b. Calculate the distance between the lines $\left( d _ { 1 } \right)$ and $\left( d _ { 2 } \right)$.
\end{enumerate}