The objective is to determine a measure of the angle between two carbon-hydrogen bonds.
A regular tetrahedron is a polyhedron whose four faces are equilateral triangles.
Electrical interactions lead to modeling the methane molecule $\mathrm{CH}_4$ as follows:

• The nuclei of hydrogen atoms occupy the positions of the four vertices of a regular tetrahedron.
• The carbon nucleus at the center of the molecule is equidistant from the four hydrogen atoms.

1. Justify that we can inscribe this tetrahedron in a cube ABCDEFGH by positioning two hydrogen atoms at vertices A and C of the cube and the two other hydrogen atoms at two other vertices of the cube. Represent the molecule in the cube given in the appendix on page 6. In the rest of the exercise, we can work in the coordinate system $(A; \overrightarrow{AB}; \overrightarrow{AD}; \overrightarrow{AE})$.
2. Prove that the carbon atom is at the center $\Omega$ of the cube.
3. Determine the approximation to the nearest tenth of a degree of the measure of the angle formed between the carbon-hydrogen bonds, that is, the angle $\widehat{A\Omega C}$.

The figure below represents a cube ABCDEFGH. The three points I, J, K are defined by the following conditions:

• I is the midpoint of segment [AD];
• J is such that $\overrightarrow{\mathrm{AJ}} = \frac{3}{4}\overrightarrow{\mathrm{AE}}$;
• K is the midpoint of segment [FG].

We place ourselves in the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$.

1. a. Give without justification the coordinates of points I, J and K. b. Determine the real numbers $a$ and $b$ such that the vector $\vec{n}(4; a; b)$ is orthogonal to the vectors $\overrightarrow{\mathrm{IJ}}$ and $\overrightarrow{\mathrm{IK}}$. c. Deduce that a Cartesian equation of the plane (IJK) is: $4x - 6y - 4z + 3 = 0$.
2. a. Give a parametric representation of the line (CG). b. Calculate the coordinates of point N, the intersection of the plane (IJK) and the line (CG). c. Place point N on the figure and construct in colour the cross-section of the cube by the plane (IJK).

In space equipped with the orthonormal coordinate system ($\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k}$) with unit 1 cm, we consider the points $\mathrm{A}$, $\mathrm{B}$, C and D with coordinates respectively $(2; 1; 4)$, $(4; -1; 0)$, $(0; 3; 2)$ and $(4; 3; -2)$.

1. Determine a parametric representation of the line (CD).
2. Let $M$ be a point on the line (CD). a. Determine the coordinates of the point $M$ such that the distance $BM$ is minimal. b. We denote H the point on the line $(\mathrm{CD})$ with coordinates $(3; 3; -1)$. Verify that the lines $(\mathrm{BH})$ and $(\mathrm{CD})$ are perpendicular. c. Show that the area of triangle BCD is equal to $12\,\mathrm{cm}^2$.
3. a. Prove that the vector $\vec{n}\begin{pmatrix}2\\1\\2\end{pmatrix}$ is a normal vector to the plane (BCD). b. Determine a Cartesian equation of the plane (BCD).

(For candidates who have not followed the specialization course)
We connect the centres of each face of a cube ABCDEFGH to form a solid IJKLMN. More precisely, the points I, J, K, L, M and N are the centres respectively of the square faces ABCD, BCGF, CDHG, ADHE, ABFE and EFGH (thus the midpoints of the diagonals of these squares).

1. Without using a coordinate system (and thus coordinates) in the reasoning, justify that the lines (IN) and (ML) are orthogonal.

In what follows, we consider the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}; \overrightarrow{\mathrm{AD}}; \overrightarrow{\mathrm{AE}})$ in which, for example, the point N has coordinates $\left(\frac{1}{2}; \frac{1}{2}; 1\right)$.

1. a. Give the coordinates of the vectors $\overrightarrow{\mathrm{NC}}$ and $\overrightarrow{\mathrm{ML}}$. b. Deduce that the lines (NC) and (ML) are orthogonal. c. From the previous questions, deduce a Cartesian equation of the plane (NCI).
2. a. Show that a Cartesian equation of the plane (NJM) is: $x - y + z = 1$. b. Is the line (DF) perpendicular to the plane (NJM)? Justify. c. Show that the intersection of the planes (NJM) and (NCI) is a line for which you will give a point and a direction vector. Name the line thus obtained using two points from the figure.

Exercise 3 — Part B
In an orthonormal coordinate system of space, consider the point $\mathrm{A}(3; 1; -5)$ and the line $d$ with parametric representation $\left\{\begin{array}{rl} x &= 2t + 1 \\ y &= -2t + 9 \\ z &= t - 3 \end{array}\right.$ where $t \in \mathbb{R}$.
1. Determine a Cartesian equation of the plane $P$ orthogonal to the line $d$ and passing through point A.
2. Show that the intersection point of plane $P$ and line $d$ is point $\mathrm{B}(5; 5; -1)$.
3. Justify that point $\mathrm{C}(7; 3; -9)$ belongs to plane $P$ then show that triangle ABC is a right isosceles triangle at A.
4. Let $t$ be a real number different from 2 and $M$ the point with parameter $t$ belonging to line $d$.
a. Justify that triangle $\mathrm{AB}M$ is right-angled.
b. Show that triangle $\mathrm{AB}M$ is isosceles at B if and only if the real number $t$ satisfies the equation $t^2 - 4t = 0$.
c. Deduce the coordinates of points $M_1$ and $M_2$ on line $d$ such that the right triangles $\mathrm{AB}M_1$ and $\mathrm{AB}M_2$ are isosceles at B.

Exercise 3 — Part C
We are given the point $\mathrm{D}(9; 1; 1)$ which is one of the two solution points from question 4.c. of Part B. The four vertices of tetrahedron ABCD are located on a sphere.
Using the results from the questions in Parts A and B above, determine the coordinates of the center of this sphere and calculate its radius.

Exercise 4 (For candidates who have not followed the specialty course)

We consider a cube $ABCDEFGH$ with edge length 1. We denote $I$ the midpoint of segment $[EF]$, $J$ the midpoint of segment $[EH]$ and $K$ the point of segment $[AD]$ such that $\overrightarrow{AK} = \frac{1}{4}\overrightarrow{AD}$. We denote $\mathscr{P}$ the plane passing through $I$ and parallel to the plane $(FHK)$.

Part A

In this part, the constructions requested will be performed without justification on the figure given in the appendix.

1. The plane $(FHK)$ intersects the line $(AE)$ at a point which we denote $M$. Construct the point $M$.
2. Construct the cross-section of the cube by the plane $\mathscr{P}$.

Part B

In this part, we equip the space with the orthonormal coordinate system $(A; \overrightarrow{AB}, \overrightarrow{AD}, \overrightarrow{AE})$. We recall that $\mathscr{P}$ is the plane passing through $I$ and parallel to the plane $(FHK)$.

1. a. Show that the vector $\vec{n}\left(\begin{array}{c} 4 \\ 4 \\ -3 \end{array}\right)$ is a normal vector to the plane $(FHK)$. b. Deduce that a Cartesian equation of the plane $(FHK)$ is: $4x + 4y - 3z - 1 = 0$. c. Determine an equation of the plane $\mathscr{P}$. d. Calculate the coordinates of the point $M'$, the point of intersection of the plane $\mathscr{P}$ and the line $(AE)$.
2. We denote $\Delta$ the line passing through point $E$ and perpendicular to the plane $\mathscr{P}$. a. Determine a parametric representation of the line $\Delta$. b. Calculate the coordinates of point $L$, the intersection of line $\Delta$ and plane $(ABC)$. c. Draw the line $\Delta$ on the figure provided in the appendix. d. Are the lines $\Delta$ and $(BF)$ intersecting? What about the lines $\Delta$ and $(CG)$? Justify.

Exercise 4 — For candidates who have not followed the speciality
5 points
On the figure given in appendix 2 to be returned with the copy:

• ABCDEFGH is a rectangular parallelepiped such that $\mathrm { AB } = 12 , \mathrm { AD } = 18$ and $\mathrm { AE } = 6$
• EBDG is a tetrahedron.

Space is referred to an orthonormal coordinate system with origin A in which the points $\mathrm { B } , \mathrm { D }$ and E have respective coordinates $\mathrm { B } ( 12 ; 0 ; 0 ) , \mathrm { D } ( 0 ; 18 ; 0 )$ and $\mathrm { E } ( 0 ; 0 ; 6 )$.

1. Prove that the plane (EBD) has the Cartesian equation $3 x + 2 y + 6 z - 36 = 0$.
2. a. Determine a parametric representation of the line (AG). b. Deduce that the line (AG) intersects the plane (EBD) at a point K with coordinates (4;6;2).
3. Is the line (AG) orthogonal to the plane (EBD)? Justify.
4. a. Let M be the midpoint of the segment $[ \mathrm { ED } ]$. Prove that the points B, K and M are collinear. b. Then construct the point K on the figure given in appendix 2 to be returned with the copy.
5. We denote by P the plane parallel to the plane (ADE) passing through the point K. a. Prove that the plane P intersects the plane (EBD) along a line parallel to the line (ED). b. Then construct on appendix 2 to be returned with the copy the intersection of the plane P and the face EBD of the tetrahedron EBDG.

Exercise 3

In the cube ABCDEFGH, we have placed the points $M$ and $N$ which are the midpoints of the segments $[ A B ]$ and $[ B C ]$ respectively. We place ourselves in the coordinate system ( $\mathrm { A } ; \overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { AE } }$ ).

1. Give without justification the coordinates of points $\mathrm { H } , \mathrm { M }$ and N.
2. We admit that the lines (CD) and (MN) are secant and we denote K their point of intersection. a. Give a parametric representation of the line (MN). We admit that a parametric representation of the line (CD) is $$\left\{ \begin{array} { l } x = t \\ y = 1 \\ z = 0 \end{array} , t \in \mathbb { R } . \right.$$ b. Determine the coordinates of point K.
3. We admit that the points $\mathrm { H } , \mathrm { M } , \mathrm { N }$ define a plane and that the line (CG) and the plane (HMN) are secant. We denote L their point of intersection. a. Verify that the vector $\vec { n } \left( \begin{array} { c } 2 \\ - 2 \\ 3 \end{array} \right)$ is a normal vector to the plane (HMN). b. Determine a Cartesian equation of the plane (HMN). c. Deduce the coordinates of point L.
4. On ANNEX 2, construct the points K and L then the cross-section of the cube ABCDEFGH by the plane (HMN).

Space is referred to an orthonormal coordinate system $( \mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k } )$. We consider the points $\mathrm { A } ( 1 ; 1 ; 4 ) ; \mathrm { B } ( 4 ; 2 ; 5 ) ; \mathrm { C } ( 3 ; 0 ; - 2 )$ and $\mathrm { J } ( 1 ; 4 ; 2 )$. We denote:

• $\mathscr { P }$ the plane passing through points $\mathrm { A }$, $\mathrm { B }$ and $\mathrm { C }$;
• $\mathscr { D }$ the line passing through point $\mathrm { J }$ and with direction vector $\overrightarrow { \mathrm { u } } \left( \begin{array} { l } 1 \\ 1 \\ 3 \end{array} \right)$.

1. Relative position of $\mathscr { P }$ and $\mathscr { D }$ a. Show that the vector $\vec { n } \left( \begin{array} { c } 1 \\ - 4 \\ 1 \end{array} \right)$ is normal to $\mathscr { P }$. b. Determine a Cartesian equation of the plane $\mathscr { P }$. c. Show that $\mathscr { D }$ is parallel to $\mathscr { P }$.

We consider the point $\mathrm { I } ( 1 ; 9 ; 0 )$ and we call $\mathscr { S }$ the sphere with center $\mathrm { I }$ and radius 6.

1. Relative position of $\mathscr { P }$ and $\mathscr { S }$ a. Show that the line $\Delta$ passing through $\mathrm { I }$ and perpendicular to the plane $\mathscr { P }$ intersects this plane $\mathscr { P }$ at the point $\mathrm { H } ( 3 ; 1 ; 2 )$. b. Calculate the distance $\mathrm { IH }$. We admit that for every point $M$ of the plane $\mathscr { P }$ we have $\mathrm { I } M \geqslant \mathrm { IH }$. c. Does the plane $\mathscr { P }$ intersect the sphere $\mathscr { S }$? Justify your answer.
2. Relative position of $\mathscr { D }$ and $\mathscr { S }$ a. Determine a parametric representation of the line $\mathscr { D }$. b. Show that a point $M$ with coordinates $( x ; y ; z )$ belongs to the sphere $\mathscr { S }$ if and only if: $$( x - 1 ) ^ { 2 } + ( y - 9 ) ^ { 2 } + z ^ { 2 } = 36 .$$ c. Show that the line $\mathscr { D }$ intersects the sphere at two distinct points.

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: $$\left\{ \begin{array} { l } { x = 1 - t } \\ { y = t } \\ { z = t } \end{array} \quad \text { where } t \in \mathbb { R } \quad \text { and } \left\{ \begin{array} { r l } x & = 1 \\ y & = t ^ { \prime } \\ z & = 1 - t ^ { \prime } \end{array} \quad \text { where } t ^ { \prime } \in \mathbb { R } . \right. \right.$$

1. Show that for every real $t$, the point $M$ belongs to the line (BH).
2. Show that the lines (BH) and (FC) are orthogonal and non-coplanar.
3. 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).

We consider a cube ABCDEFGH. The point I is the midpoint of segment $[\mathrm{EF}]$, the point J is the midpoint of segment [BC] and the point K is the midpoint of segment [AE].

1. Are the lines $(\mathrm{AI})$ and $(\mathrm{KH})$ parallel? Justify your answer.

In the following, we place ourselves in the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$.
2. a. Give the coordinates of points I and J. b. Show that the vectors $\overrightarrow{\mathrm{IJ}}, \overrightarrow{\mathrm{AE}}$ and $\overrightarrow{\mathrm{AC}}$ are coplanar.
We consider the plane $\mathscr{P}$ with equation $x + 3y - 2z + 2 = 0$ as well as the lines $d_1$ and $d_2$ defined by the parametric representations below:
$$d_{1} : \left\{ \begin{array}{rl} x & = 3 + t \\ y & = 8 - 2t \\ z & = -2 + 3t \end{array} , t \in \mathbb{R} \text{ and } d_{2} : \left\{ \begin{array}{rl} x & = 4 + t \\ y & = 1 + t \\ z & = 8 + 2t \end{array} , t \in \mathbb{R}. \right. \right.$$

1. Are the lines $d_1$ and $d_2$ parallel? Justify your answer.
2. Show that the line $d_2$ is parallel to the plane $\mathscr{P}$.
3. Show that the point $\mathrm{L}(4 ; 0 ; 3)$ is the orthogonal projection of the point $\mathrm{M}(5 ; 3 ; 1)$ onto the plane $\mathscr{P}$.

We consider a cube ABCDEFGH with edge 8 cm and centre $\Omega$.
The points P, Q and R are defined by $\overrightarrow{AP} = \frac{3}{4}\overrightarrow{AB}$, $\overrightarrow{AQ} = \frac{3}{4}\overrightarrow{AE}$ and $\overrightarrow{FR} = \frac{1}{4}\overrightarrow{FG}$. We use the orthonormal coordinate system $(A; \vec{\imath}, \vec{\jmath}, \vec{k})$ with: $\vec{\imath} = \frac{1}{8}\overrightarrow{AB}$, $\vec{\jmath} = \frac{1}{8}\overrightarrow{AD}$ and $\vec{k} = \frac{1}{8}\overrightarrow{AE}$.

Part I

1. In this coordinate system, we admit that the coordinates of point R are $(8; 2; 8)$. Give the coordinates of points P and Q.
2. Show that the vector $\vec{n}(1; -5; 1)$ is a normal vector to the plane (PQR).
3. Justify that a Cartesian equation of the plane (PQR) is $x - 5y + z - 6 = 0$.

Part II

We denote L the orthogonal projection of point $\Omega$ onto the plane (PQR).

1. Justify that the coordinates of point $\Omega$ are $(4; 4; 4)$.
2. Give a parametric representation of the line $d$ perpendicular to the plane (PQR) and passing through $\Omega$.
3. Show that the coordinates of point L are $\left(\frac{14}{3}; \frac{2}{3}; \frac{14}{3}\right)$.
4. Calculate the distance from point $\Omega$ to the plane (PQR).

Consider a rectangular parallelepiped ABCDEFGH such that $\mathrm{AB} = \mathrm{AD} = 1$ and $\mathrm{AE} = 2$. Point I is the midpoint of segment [AE]. Point K is the midpoint of segment [DC]. Point L is defined by: $\overrightarrow{\mathrm{DL}} = \frac{3}{2}\overrightarrow{\mathrm{AI}}$. N is the orthogonal projection of point D onto the plane (AKL).
We use the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AI}})$. We admit that point L has coordinates $\left(0; 1; \frac{3}{2}\right)$.

1. Determine the coordinates of vectors $\overrightarrow{\mathrm{AK}}$ and $\overrightarrow{\mathrm{AL}}$.
2. a. Prove that the vector $\vec{n}$ with coordinates $(6; -3; 2)$ is a normal vector to the plane (AKL). b. Deduce a Cartesian equation of the plane (AKL). c. Determine a system of parametric equations of the line $\Delta$ passing through D and perpendicular to the plane (AKL). d. Deduce that the point N with coordinates $\left(\frac{18}{49}; \frac{40}{49}; \frac{6}{49}\right)$ is the orthogonal projection of point D onto the plane (AKL).

We recall that the volume $\mathcal{V}$ of a tetrahedron is given by the formula: $$\mathcal{V} = \frac{1}{3} \times (\text{area of the base}) \times \text{height.}$$

1. a. Calculate the volume of tetrahedron ADKL using triangle ADK as the base. b. Calculate the distance from point D to the plane (AKL). c. Deduce from the previous questions the area of triangle AKL.

In an orthonormal coordinate system of space, we consider the following points: $$\mathrm{A}(2;-1;0),\quad \mathrm{B}(3;-1;2),\quad \mathrm{C}(0;4;1)\quad \text{and}\quad \mathrm{S}(0;1;4).$$

1. Show that triangle ABC is right-angled at A.
2. a. Show that the vector $\vec{n}\begin{pmatrix}2\\1\\-1\end{pmatrix}$ is orthogonal to the plane (ABC). b. Deduce a Cartesian equation of the plane (ABC). c. Show that the points A, B, C and S are not coplanar.
3. Let (d) be the line perpendicular to the plane (ABC) passing through S. It intersects the plane (ABC) at H. a. Determine a parametric representation of the line (d). b. Show that the coordinates of point H are $\mathrm{H}(2;2;3)$.
4. We recall that the volume $V$ of a tetrahedron is $V = \dfrac{\text{area of base} \times \text{height}}{3}$. Calculate the volume of tetrahedron SABC.
5. a. Calculate the length SA. b. We are told that $\mathrm{SB} = \sqrt{17}$. Deduce an approximate measure of the angle $\widehat{\mathrm{ASB}}$ to the nearest tenth of a degree.

Main topics covered: Space geometry with respect to an orthonormal coordinate system; orthogonality in space
In an orthonormal coordinate system $( \mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k } )$ we consider

• the point A with coordinates $( 1 ; 3 ; 2 )$,
• the vector $\vec { u }$ with coordinates $\left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right)$
• the line $d$ passing through the origin O of the coordinate system and having $\vec { u }$ as its direction vector.

The purpose of this exercise is to determine the point on $d$ closest to point A and to study some properties of this point.

1. Determine a parametric representation of the line $d$.
2. Let $t$ be any real number, and $M$ a point on the line $d$, the point $M$ having coordinates $( t ; t ; 0 )$. a. We denote $AM$ the distance between points A and $M$. Prove that: $$AM ^ { 2 } = 2 t ^ { 2 } - 8 t + 14 .$$ b. Prove that the point $M _ { 0 }$ with coordinates $( 2 ; 2 ; 0 )$ is the point on the line $d$ for which the distance $AM$ is minimal. We will assume that the distance $AM$ is minimal when its square $AM ^ { 2 }$ is minimal.
3. Prove that the lines $( A M _ { 0 } )$ and $d$ are orthogonal.
4. We call $A ^ { \prime }$ the orthogonal projection of point $A$ onto the plane with Cartesian equation $z = 0$. The point $A ^ { \prime }$ therefore has coordinates $( 1 ; 3 ; 0 )$.
   Prove that the point $M _ { 0 }$ is the point of the plane $\left( A A ^ { \prime } M _ { 0 } \right)$ closest to point O, the origin of the coordinate system.
5. Calculate the volume of the pyramid $O M _ { 0 } A ^ { \prime } A$.
   Recall that the volume of a pyramid is given by: $V = \frac { 1 } { 3 } \mathscr { B } h$, where $\mathscr { B }$ is the area of a base and $h$ is the height of the pyramid corresponding to this base.

We consider the cube ABCDEFGH. We are given three points I, J and K satisfying: $$\overrightarrow { \mathrm { EI } } = \frac { 1 } { 4 } \overrightarrow { \mathrm { EH } } , \quad \overrightarrow { \mathrm { EJ } } = \frac { 1 } { 4 } \overrightarrow { \mathrm { EF } } , \quad \overrightarrow { \mathrm { BK } } = \frac { 1 } { 4 } \overrightarrow { \mathrm { BF } }$$ We use the orthonormal coordinate system $(A ; \overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { AE } })$.

1. Give without justification the coordinates of points I, J and K.
2. Prove that the vector $\overrightarrow { \mathrm { AG } }$ is normal to the plane (IJK).
3. Show that a Cartesian equation of the plane (IJK) is $4 x + 4 y + 4 z - 5 = 0$.
4. Determine a parametric representation of the line (BC).
5. Deduce the coordinates of point L, the point of intersection of the line (BC) with the plane (IJK).
6. On the figure in the appendix, place point L and construct the intersection of the plane (IJK) with the face (BCGF).
7. Let $\mathrm { M } \left( \frac { 1 } { 4 } ; 1 ; 0 \right)$. Show that the points I, J, L and M are coplanar.

In space with respect to an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider the points: A with coordinates $(2; 0; 0)$, B with coordinates $(0; 3; 0)$ and C with coordinates $(0; 0; 1)$.
The objective of this exercise is to calculate the area of triangle ABC.

1. a. Show that the vector $\vec{n}\left(\begin{array}{l}3\\2\\6\end{array}\right)$ is normal to the plane (ABC). b. Deduce that a Cartesian equation of the plane (ABC) is: $3x + 2y + 6z - 6 = 0$.
2. We denote by $d$ the line passing through O and perpendicular to the plane (ABC). a. Determine a parametric representation of the line $d$. b. Show that the line $d$ intersects the plane (ABC) at the point H with coordinates $\left(\frac{18}{49}; \frac{12}{49}; \frac{36}{49}\right)$. c. Calculate the distance OH.
3. We recall that the volume of a pyramid is given by: $V = \frac{1}{3}\mathscr{B}h$, where $\mathscr{B}$ is the area of a base and $h$ is the height of the pyramid corresponding to this base. By calculating in two different ways the volume of the pyramid OABC, determine the area of triangle ABC.

In space, consider the cube ABCDEFGH with edge length equal to 1. We equip the space with the orthonormal coordinate system (A ; $\overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { AE } }$). Consider the point M such that $\overrightarrow { \mathrm { BM } } = \frac { 1 } { 3 } \overrightarrow { \mathrm { BH } }$.

1. By reading the graph, give the coordinates of points $\mathrm { B } , \mathrm { D } , \mathrm { E } , \mathrm { G }$ and H.
2. a. What is the nature of triangle EGD? Justify your answer. b. It is admitted that the area of an equilateral triangle with side $c$ is equal to $\frac { \sqrt { 3 } } { 4 } c ^ { 2 }$. Show that the area of triangle EGD is equal to $\frac { \sqrt { 3 } } { 2 }$.
3. Prove that the coordinates of M are $\left( \frac { 2 } { 3 } ; \frac { 1 } { 3 } ; \frac { 1 } { 3 } \right)$.
4. a. Justify that the vector $\vec { n } ( - 1 ; 1 ; 1 )$ is normal to the plane (EGD). b. Deduce that a Cartesian equation of the plane (EGD) is: $- x + y + z - 1 = 0$. c. Let $\mathscr { D }$ be the line perpendicular to the plane (EGD) and passing through point M. Show that a parametric representation of this line is: $$\mathscr { D } : \left\{ \begin{aligned} x & = \frac { 2 } { 3 } - t \\ y & = \frac { 1 } { 3 } + t , t \in \mathbb { R } \\ z & = \frac { 1 } { 3 } + t \end{aligned} \right.$$
5. The purpose of this question is to calculate the volume of the pyramid GEDM. a. Let K be the foot of the height of the pyramid GEDM from point M. Prove that the coordinates of point K are $\left( \frac { 1 } { 3 } ; \frac { 2 } { 3 } ; \frac { 2 } { 3 } \right)$. b. Deduce the volume of the pyramid GEDM. Recall that the volume $V$ of a pyramid is given by the formula $V = \frac { b \times h } { 3 }$ where $b$ denotes the area of a base and h the associated height.

In space equipped with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ) with unit 1 cm, we consider the following points:
$$\mathrm { J } ( 2 ; 0 ; 1 ) , \quad \mathrm { K } ( 1 ; 2 ; 1 ) \text { and } \quad \mathrm { L } ( - 2 ; - 2 ; - 2 )$$

1. a. Show that triangle JKL is right-angled at J. b. Calculate the exact value of the area of triangle JKL in $\mathrm { cm } ^ { 2 }$. c. Determine an approximate value to the nearest tenth of the geometric angle $\widehat { \mathrm { JKL } }$.
   
2. a. Prove that the vector $\vec { n }$ with coordinates $\left( \begin{array} { c } 6 \\ 3 \\ - 10 \end{array} \right)$ is a normal vector to the plane (JKL). b. Deduce a Cartesian equation of the plane (JKL).

In the following, T denotes the point with coordinates ( $10 ; 9 ; - 6$ ).
3. a. Determine a parametric representation of the line $\Delta$ perpendicular to the plane (JKL) and passing through T. b. Determine the coordinates of point H, the orthogonal projection of point T onto the plane (JKL). c. We recall that the volume $V$ of a tetrahedron is given by the formula:
$$V = \frac { 1 } { 3 } \mathscr { B } \times h \text { where } \mathscr { B } \text { denotes the area of a base and } h \text { the corresponding height }$$
Calculate the exact value of the volume of tetrahedron JKLT in $\mathrm { cm } ^ { 3 }$.

Exercise 3 (7 points) Theme: geometry in space An exhibition of contemporary art takes place in a room in the shape of a rectangular parallelepiped with width 6 m, length 8 m and height 4 m. It is represented by the rectangular parallelepiped OBCDEFGH where $\mathrm { OB } = 6 \mathrm {~m} , \mathrm { OD } = 8 \mathrm {~m}$ and $\mathrm { OE } = 4 \mathrm {~m}$. We use the orthonormal coordinate system $( \mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k } )$ such that $\vec { \imath } = \frac { 1 } { 6 } \overrightarrow { \mathrm { OB } } , \vec { \jmath } = \frac { 1 } { 8 } \overrightarrow { \mathrm { OD } }$ and $\vec { k } = \frac { 1 } { 4 } \overrightarrow { \mathrm { OE } }$. In this coordinate system we have, in particular $\mathrm { C } ( 6 ; 8 ; 0 ) , \mathrm { F } ( 6 ; 0 ; 4 )$ and $\mathrm { G } ( 6 ; 8 ; 4 )$. One of the exhibited works is a glass triangle represented by triangle ART which has vertices $\mathrm { A } ( 6 ; 0 ; 2 )$, $\mathrm { R } ( 6 ; 3 ; 4 )$ and $\mathrm { T } ( 3 ; 0 ; 4 )$. Finally, S is the point with coordinates $\left( 3 ; \frac { 5 } { 2 } ; 0 \right)$.

1. a. Verify that triangle ART is isosceles with apex A. b. Calculate the dot product $\overrightarrow { \mathrm { AR } } \cdot \overrightarrow { \mathrm { AT } }$. c. Deduce an approximate value to 0.1 degree of the angle $\widehat { \mathrm { RAT } }$.
2. a. Justify that the vector $\vec { n } \left( \begin{array} { c } 2 \\ - 2 \\ 3 \end{array} \right)$ is a normal vector to the plane (ART). b. Deduce a Cartesian equation of the plane (ART).
3. A laser beam directed towards triangle ART is emitted from the floor from point S. It is admitted that this beam is perpendicular to the plane (ART). a. Let $\Delta$ be the line perpendicular to the plane (ART) and passing through point S. Justify that the system below is a parametric representation of the line $\Delta$: $$\left\{ \begin{aligned} x & = 3 + 2 k \\ y & = \frac { 5 } { 2 } - 2 k , \text { with } k \in \mathbb { R } . \\ z & = 3 k \end{aligned} \right.$$ b. Let L be the point of intersection of the line $\Delta$ with the plane (ART). Prove that L has coordinates $\left( 5 ; \frac { 1 } { 2 } ; 3 \right)$.
4. The artist installs a rail represented by the segment [DK] where K is the midpoint of segment [EH]. On this rail, he positions a laser light source at a point N of segment [DK] and directs this second laser beam towards point S. a. Show that, for every real $t$ in the interval $[ 0 ; 1 ]$, the point N with coordinates $( 0 ; 8 - 4 t ; 4 t )$ is a point of segment [DK]. b. Calculate the exact coordinates of point N such that the two laser beams represented by segments [SL] and [SN] are perpendicular.

Exercise 4 Geometry in space
In the figure below, ABCDEFGH is a rectangular parallelepiped such that $\mathrm{AB} = 5$, $\mathrm{AD} = 3$ and $\mathrm{AE} = 2$. The space is equipped with an orthonormal coordinate system with origin A in which the points B, D and E have coordinates respectively $(5; 0; 0)$, $(0; 3; 0)$ and $(0; 0; 2)$.

1. a. Give, in the coordinate system considered, the coordinates of points H and G. b. Give a parametric representation of the line (GH).
2. Let M be a point of the segment $[\mathrm{GH}]$ such that $\overrightarrow{\mathrm{HM}} = k\overrightarrow{\mathrm{HG}}$ with $k$ a real number in the interval $[0; 1]$. a. Justify that the coordinates of M are $(5k; 3; 2)$. b. Deduce from this that $\overrightarrow{\mathrm{AM}} \cdot \overrightarrow{\mathrm{CM}} = 25k^{2} - 25k + 4$. c. Determine the values of $k$ for which AMC is a triangle right-angled at M.

For the rest of the exercise, we consider that point M has coordinates $(1; 3; 2)$. We admit that triangle AMC is right-angled at M. We recall that the volume of a tetrahedron is given by the formula $\frac{1}{3} \times$ Area of the base $\times h$ where $h$ is the height relative to the base.

1. We consider the point K with coordinates $(1; 3; 0)$. a. Determine a Cartesian equation of the plane (ACD). b. Justify that point K is the orthogonal projection of point M onto the plane (ACD). c. Deduce from this the volume of the tetrahedron MACD.
2. We denote P the orthogonal projection of point D onto the plane (AMC). Calculate the distance DP; give a value rounded to $10^{-1}$.

Exercise 4: Geometry in Space
In space equipped with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider the points $$\mathrm{A}(0; 8; 6), \quad \mathrm{B}(6; 4; 4) \quad \text{and} \quad \mathrm{C}(2; 4; 0).$$

1. a. Justify that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are not collinear. b. Show that the vector $\vec{n}(1; 2; -1)$ is a normal vector to the plane (ABC). c. Determine a Cartesian equation of the plane (ABC).
2. Let D and E be the points with respective coordinates $(0; 0; 6)$ and $(6; 6; 0)$. a. Determine a parametric representation of the line (DE). b. Show that the midpoint I of the segment [BC] belongs to the line (DE).
3. We consider the triangle ABC. a. Determine the nature of triangle ABC. b. Calculate the area of triangle ABC in square units. c. Calculate $\overrightarrow{\mathrm{AB}} \cdot \overrightarrow{\mathrm{AC}}$. d. Deduce a measure of the angle $\widehat{\mathrm{BAC}}$ rounded to 0.1 degree.
4. We consider the point H with coordinates $\left(\dfrac{5}{3}; \dfrac{10}{3}; -\dfrac{5}{3}\right)$. Show that $H$ is the orthogonal projection of the point $O$ onto the plane (ABC). Deduce the distance from point O to the plane (ABC).

The solid ABCDEFGH is a cube. We place ourselves in the orthonormal coordinate system (A ; $\vec { \imath } , \vec { \jmath } , \vec { k }$) of space in which the coordinates of points B, D and E are: $$\mathrm { B } ( 3 ; 0 ; 0 ) , \quad \mathrm { D } ( 0 ; 3 ; 0 ) \quad \text { and } \quad \mathrm { E } ( 0 ; 0 ; 3 ) .$$ We consider the points $\mathrm { P } ( 0 ; 0 ; 1 ) , \quad \mathrm { Q } ( 0 ; 2 ; 3 )$ and $\mathrm { R } ( 1 ; 0 ; 3 )$.

1. Place the points P, Q and R on the figure in the APPENDIX which must be returned with your work.
2. Show that the triangle PQR is isosceles at R.
3. Justify that the points P, Q and R define a plane.
4. We are now interested in the distance between point E and the plane (PQR). a. Show that the vector $\vec { u } ( 2 ; 1 ; - 1 )$ is normal to the plane (PQR). b. Deduce a Cartesian equation of the plane (PQR). c. Determine a parametric representation of the line (d) passing through point E and orthogonal to the plane (PQR). d. Show that the point $\mathrm { L } \left( \frac { 2 } { 3 } ; \frac { 1 } { 3 } ; \frac { 8 } { 3 } \right)$ is the orthogonal projection of point E onto the plane (PQR). e. Determine the distance between point E and the plane (PQR).
5. By choosing the triangle EQR as the base, show that the volume of the tetrahedron EPQR is $\frac { 2 } { 3 }$. We recall that the volume $V$ of a tetrahedron is given by the formula: $$V = \frac { 1 } { 3 } \times \text { area of a base } \times \text { corresponding height. }$$
6. Find, using the two previous questions, the area of triangle PQR.

In an orthonormal coordinate system $( \mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k } )$ of space, we consider the points
$$\mathrm { A } ( - 3 ; 1 ; 3 ) , \mathrm { B } ( 2 ; 2 ; 3 ) , \mathrm { C } ( 1 ; 7 ; - 1 ) , \mathrm { D } ( - 4 ; 6 ; - 1 ) \text { and K(-3;14;14). }$$

1. a. Calculate the coordinates of the vectors $\overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { DC } }$ and $\overrightarrow { \mathrm { AD } }$. b. Show that the quadrilateral ABCD is a rectangle. c. Calculate the area of rectangle ABCD.
2. a. Justify that the points $\mathrm { A } , \mathrm { B }$ and D define a plane. b. Show that the vector $\vec { n } ( - 2 ; 10 ; 13 )$ is a normal vector to the plane (ABD). c. Deduce a Cartesian equation of the plane (ABD).
3. a. Give a parametric representation of the line $\Delta$ orthogonal to the plane (ABD) and passing through point K. b. Determine the coordinates of point I, the orthogonal projection of point K onto the plane (ABD). c. Show that the height of the pyramid KABCD with base ABCD and apex K equals $\sqrt { 273 }$.
4. Calculate the volume $V$ of the pyramid KABCD.

Recall that the volume V of a pyramid is given by the formula:
$$V = \frac { 1 } { 3 } \times \text { base area × height. }$$