Topics: Geometry in space In space with respect to an orthonormal coordinate system $(\mathrm { O } , \vec { \imath } , \vec { \jmath } , \vec { k })$, we consider:
the line $\mathscr { D }$ passing through the point $\mathrm { A } ( 2 ; 4 ; 0 )$ and whose direction vector is $\vec { u } \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right)$;
the line $\mathscr { D } ^ { \prime }$ whose parametric representation is: $\left\{ \begin{array} { r l } x & = 3 \\ y & = 3 + t \\ z & = 3 + t \end{array} , t \in \mathbb { R } \right.$.
a. Give the coordinates of a direction vector $\overrightarrow { u ^ { \prime } }$ of the line $\mathscr { D } ^ { \prime }$. b. Show that the lines $\mathscr { D }$ and $\mathscr { D } ^ { \prime }$ are not parallel. c. Determine a parametric representation of the line $\mathscr { D }$.
We admit in the rest of this exercise that there exists a unique line $\Delta$ perpendicular to the lines $\mathscr { D }$ and $\mathscr { D } ^ { \prime }$. This line $\Delta$ intersects each of the lines $\mathscr { D }$ and $\mathscr { D } ^ { \prime }$. We will call M the intersection point of $\Delta$ and $\mathscr { D }$, and $\mathrm { M } ^ { \prime }$ the intersection point of $\Delta$ and $\mathscr { D } ^ { \prime }$. We propose to determine the distance $\mathrm { MM } ^ { \prime }$ called the ``distance between the lines $\mathscr { D }$ and $\mathscr { D } ^ { \prime }$''.
Show that the vector $\vec { v } \left( \begin{array} { c } 2 \\ - 1 \\ 1 \end{array} \right)$ is a direction vector of the line $\Delta$.
We denote by $\mathscr { P }$ the plane containing the lines $\mathscr { D }$ and $\Delta$, that is, the plane passing through point A and with direction vectors $\vec { u }$ and $\vec { v }$. a. Show that the vector $\vec { n } \left( \begin{array} { c } 2 \\ - 1 \\ - 5 \end{array} \right)$ is a normal vector to the plane $\mathscr { P }$. b. Deduce that an equation of the plane $\mathscr { P }$ is: $2 x - y - 5 z = 0$. c. We recall that $\mathrm { M } ^ { \prime }$ is the intersection point of the lines $\Delta$ and $\mathscr { D } ^ { \prime }$. Justify that $\mathrm { M } ^ { \prime }$ is also the intersection point of $\mathscr { D } ^ { \prime }$ and the plane $\mathscr { P }$. Deduce that the coordinates of point $\mathrm { M } ^ { \prime }$ are $( 3 ; 1 ; 1 )$.
a. Determine a parametric representation of the line $\Delta$. b. Justify that point M has coordinates $( 1 ; 2 ; 0 )$. c. Calculate the distance $\mathrm { MM } ^ { \prime }$.
We consider the line $d$ with parametric representation $\left\{ \begin{aligned} x & = 5 t \\ y & = 2 + 5 t \\ z & = 1 + t \end{aligned} \right.$ with $t \in \mathbb { R }$. a. Show that the line $d$ is parallel to the plane $\mathscr { P }$. b. We denote by $\ell$ the distance from a point N of the line $d$ to the plane $\mathscr { P }$. Express the volume of the tetrahedron $\mathrm { ANMM } ^ { \prime }$ as a function of $\ell$. We recall that the volume of a tetrahedron is given by: $V = \frac { 1 } { 3 } \times B \times h$ where $B$ denotes the area of a base and $h$ the height relative to this base. c. Justify that, if $\mathrm { N } _ { 1 }$ and $\mathrm { N } _ { 2 }$ are any two points of the line $d$, the tetrahedra $A N _ { 1 } M M ^ { \prime }$ and $A N _ { 2 } M M ^ { \prime }$ have the same volume.
\section*{Exercise 4 — 7 points}
Topics: Geometry in space\\
In space with respect to an orthonormal coordinate system $(\mathrm { O } , \vec { \imath } , \vec { \jmath } , \vec { k })$, we consider:
\begin{itemize}
\item the line $\mathscr { D }$ passing through the point $\mathrm { A } ( 2 ; 4 ; 0 )$ and whose direction vector is $\vec { u } \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right)$;
\item the line $\mathscr { D } ^ { \prime }$ whose parametric representation is: $\left\{ \begin{array} { r l } x & = 3 \\ y & = 3 + t \\ z & = 3 + t \end{array} , t \in \mathbb { R } \right.$.
\end{itemize}
\begin{enumerate}
\item a. Give the coordinates of a direction vector $\overrightarrow { u ^ { \prime } }$ of the line $\mathscr { D } ^ { \prime }$.\\
b. Show that the lines $\mathscr { D }$ and $\mathscr { D } ^ { \prime }$ are not parallel.\\
c. Determine a parametric representation of the line $\mathscr { D }$.
\end{enumerate}
We admit in the rest of this exercise that there exists a unique line $\Delta$ perpendicular to the lines $\mathscr { D }$ and $\mathscr { D } ^ { \prime }$. This line $\Delta$ intersects each of the lines $\mathscr { D }$ and $\mathscr { D } ^ { \prime }$. We will call M the intersection point of $\Delta$ and $\mathscr { D }$, and $\mathrm { M } ^ { \prime }$ the intersection point of $\Delta$ and $\mathscr { D } ^ { \prime }$.\\
We propose to determine the distance $\mathrm { MM } ^ { \prime }$ called the ``distance between the lines $\mathscr { D }$ and $\mathscr { D } ^ { \prime }$''.
\begin{enumerate}
\setcounter{enumi}{1}
\item Show that the vector $\vec { v } \left( \begin{array} { c } 2 \\ - 1 \\ 1 \end{array} \right)$ is a direction vector of the line $\Delta$.
\item We denote by $\mathscr { P }$ the plane containing the lines $\mathscr { D }$ and $\Delta$, that is, the plane passing through point A and with direction vectors $\vec { u }$ and $\vec { v }$.\\
a. Show that the vector $\vec { n } \left( \begin{array} { c } 2 \\ - 1 \\ - 5 \end{array} \right)$ is a normal vector to the plane $\mathscr { P }$.\\
b. Deduce that an equation of the plane $\mathscr { P }$ is: $2 x - y - 5 z = 0$.\\
c. We recall that $\mathrm { M } ^ { \prime }$ is the intersection point of the lines $\Delta$ and $\mathscr { D } ^ { \prime }$.\\
Justify that $\mathrm { M } ^ { \prime }$ is also the intersection point of $\mathscr { D } ^ { \prime }$ and the plane $\mathscr { P }$.\\
Deduce that the coordinates of point $\mathrm { M } ^ { \prime }$ are $( 3 ; 1 ; 1 )$.
\item a. Determine a parametric representation of the line $\Delta$.\\
b. Justify that point M has coordinates $( 1 ; 2 ; 0 )$.\\
c. Calculate the distance $\mathrm { MM } ^ { \prime }$.
\item We consider the line $d$ with parametric representation $\left\{ \begin{aligned} x & = 5 t \\ y & = 2 + 5 t \\ z & = 1 + t \end{aligned} \right.$ with $t \in \mathbb { R }$.\\
a. Show that the line $d$ is parallel to the plane $\mathscr { P }$.\\
b. We denote by $\ell$ the distance from a point N of the line $d$ to the plane $\mathscr { P }$.\\
Express the volume of the tetrahedron $\mathrm { ANMM } ^ { \prime }$ as a function of $\ell$.\\
We recall that the volume of a tetrahedron is given by: $V = \frac { 1 } { 3 } \times B \times h$ where $B$ denotes the area of a base and $h$ the height relative to this base.\\
c. Justify that, if $\mathrm { N } _ { 1 }$ and $\mathrm { N } _ { 2 }$ are any two points of the line $d$, the tetrahedra $A N _ { 1 } M M ^ { \prime }$ and $A N _ { 2 } M M ^ { \prime }$ have the same volume.
\end{enumerate}