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.

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).

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.

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.

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.

This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. A correct answer earns one point. An incorrect answer, multiple answers, or no answer to a question earns or loses no points.
In space with respect to an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider the points $A(1; 0; 2)$, $B(2; 1; 0)$, $C(0; 1; 2)$ and the line $\Delta$ whose parametric representation is: $$\left\{\begin{array}{rl}x & = 1 + 2t \\ y & = -2 + t \\ z & = 4 - t\end{array}, t \in \mathbb{R}\right.$$

1. Which of the following points belongs to the line $\Delta$?
   Answer A: $M(2; 1; -1)$; Answer B: $N(-3; -4; 6)$; Answer C: $P(-3; -4; 2)$; Answer D: $Q(-5; -5; 1)$.
   
2. The vector $\overrightarrow{AB}$ has coordinates:
   $$\begin{array}{ll} \text{Answer A}: \left(\begin{array}{c} 1.5 \\ 0.5 \\ 1 \end{array}\right); & \text{Answer B}: \left(\begin{array}{c} -1 \\ -1 \\ 2 \end{array}\right); \\ \text{Answer C}: \left(\begin{array}{c} 1 \\ 1 \\ -2 \end{array}\right) & \text{Answer D}: \left(\begin{array}{l} 3 \\ 1 \\ 2 \end{array}\right). \end{array}$$
   
3. A parametric representation of the line (AB) is:
   $$\begin{array}{ll} \text{Answer A}: \left\{\begin{array}{l} x = 1 + 2t \\ y = t \\ z = 2 \end{array}, t \in \mathbb{R}\right. & \text{Answer B}: \left\{\begin{array}{l} x = 2 - t \\ y = 1 - t \\ z = 2t \end{array}, t \in \mathbb{R}\right. \\ \text{Answer C}: \left\{\begin{array}{l} x = 2 + t \\ y = 1 + t \\ z = 2t \end{array}, t \in \mathbb{R}\right. & \text{Answer D}: \left\{\begin{array}{l} x = 1 + t \\ y = 1 + t \\ z = 2 - 2t \end{array}, t \in \mathbb{R}\right. \end{array}$$
   
4. A Cartesian equation of the plane passing through point C and orthogonal to the line $\Delta$ is: Answer A: $x - 2y + 4z - 6 = 0$; Answer B: $2x + y - z + 1 = 0$; Answer C: $2x + y - z - 1 = 0$; Answer D: $y + 2z - 5 = 0$.
   
5. We consider the point D defined by the vector relation $\overrightarrow{OD} = 3\overrightarrow{OA} - \overrightarrow{OB} - \overrightarrow{OC}$.
   Answer A: $\overrightarrow{AD}$, $\overrightarrow{AB}$, $\overrightarrow{AC}$ are coplanar; Answer B: $\overrightarrow{AD} = \overrightarrow{BC}$; Answer C: D has coordinates $(3; -1; -1)$; Answer D: the points A, B, C and D are collinear.

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 $\mathscr{P}$.

This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct.
A correct answer earns one point. An incorrect answer, a multiple answer, or the absence of an answer to a question earns neither points nor deducts points. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.
SABCD is a regular pyramid with square base ABCD in which all edges have the same length. The point I is the center of the square ABCD. We assume that: $\mathrm{IC} = \mathrm{IB} = \mathrm{IS} = 1$. The points K, L and M are the midpoints of edges [SD], [SC] and [SB] respectively.

1. The following lines are not coplanar: a. (DK) and (SD) b. (AS) and (IC) c. (AC) and (SB) d. (LM) and (AD)

For the following questions, we place ourselves in the orthonormal coordinate system of space $(\mathrm{I}; \overrightarrow{\mathrm{IC}}, \overrightarrow{\mathrm{IB}}, \overrightarrow{\mathrm{IS}})$. In this coordinate system, we are given the coordinates of the following points: $$\mathrm{I}(0;0;0) \quad ; \quad \mathrm{A}(-1;0;0) \quad ; \quad \mathrm{B}(0;1;0) \quad ; \quad \mathrm{C}(1;0;0) \quad ; \quad \mathrm{D}(0;-1;0) \quad ; \quad \mathrm{S}(0;0;1).$$

1. The coordinates of the midpoint N of [KL] are: a. $\left(\frac{1}{4};\frac{1}{4};\frac{1}{4}\right)$ b. $\left(\frac{1}{4};-\frac{1}{4};\frac{1}{2}\right)$ c. $\left(-\frac{1}{4};\frac{1}{4};\frac{1}{2}\right)$ d. $\left(-\frac{1}{2};\frac{1}{2};1\right)$
2. The coordinates of the vector $\overrightarrow{\mathrm{AS}}$ are: a. $\left(\begin{array}{l}1\\1\\0\end{array}\right)$ b. $\left(\begin{array}{l}1\\0\\1\end{array}\right)$ c. $\left(\begin{array}{c}2\\1\\-1\end{array}\right)$ d. $\left(\begin{array}{l}1\\1\\1\end{array}\right)$
3. A parametric representation of the line (AS) is: a. $\left\{\begin{array}{rl}x &= -1-t\\y &= t\\z &= -t\end{array}\right.$ b. $\left\{\begin{aligned}x =& -1+2t\\y =& 0\\z =& 1+2t\end{aligned}\right.$ c. $\left\{\begin{aligned}x &= t\\y &= 0\\z &= 1+t\end{aligned}\right.$ d. $\left\{\begin{aligned}x &= -1-t\\y &= 1+t\\z &= 1-t\end{aligned}\right. (t \in \mathbb{R})$
4. A Cartesian equation of the plane (SCB) is: a. $y+z-1=0$ b. $x+y+z-1=0$ c. $x-y+z=0$ d. $x+z-1=0$

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 with respect to an orthonormal coordinate system $( \mathrm { O } , \vec { \imath } , \vec { \jmath } , \vec { k } )$, we consider the points $$\mathrm { A } ( - 1 ; - 1 ; 3 ) , \quad \mathrm { B } ( 1 ; 1 ; 2 ) , \quad \mathrm { C } ( 1 ; - 1 ; 7 )$$ We also consider the line $\Delta$ passing through the points $\mathrm { D } ( - 1 ; 6 ; 8 )$ and $\mathrm { E } ( 11 ; - 9 ; 2 )$.

1. a. Verify that the line $\Delta$ admits the following parametric representation: $$\left\{ \begin{aligned} x & = - 1 + 4 t \\ y & = 6 - 5 t \quad \text { with } t \in \mathbb { R } \\ z & = 8 - 2 t \end{aligned} \right.$$ b. Specify a parametric representation of the line $\Delta ^ { \prime }$ parallel to $\Delta$ and passing through the origin O of the coordinate system. c. Does the point $\mathrm { F } ( 1.36 ; - 1.7 ; - 0.7 )$ belong to the line $\Delta ^ { \prime }$?
2. a. Show that the points $\mathrm { A }$, $\mathrm { B }$ and $\mathrm { C }$ define a plane. b. Show that the line $\Delta$ is perpendicular to the plane (ABC). c. Show that a Cartesian equation of the plane (ABC) is: $4 x - 5 y - 2 z + 5 = 0$.
3. a. Show that the point $\mathrm { G } ( 7 ; - 4 ; 4 )$ belongs to the line $\Delta$. b. Determine the coordinates of the point H, the orthogonal projection of point G onto the plane (ABC). c. Deduce that the distance from point G to the plane (ABC) is equal to $3 \sqrt { 5 }$.
4. a. Show that the triangle ABC is right-angled at A. b. Calculate the volume $V$ of the tetrahedron ABCG. We recall that the volume $V$ of a tetrahedron is given by the formula $V = \frac { 1 } { 3 } \times B \times h$ where B is the area of a base and h the height corresponding to this base.

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:
$$V = \frac { 1 } { 3 } \times \text { base area × height. }$$

Exercise 2 — 7 points
Theme: Geometry in space
In space, referred to an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider the points: $$\mathrm{A}(2; 0; 3),\ \mathrm{B}(0; 2; 1),\ \mathrm{C}(-1; -1; 2)\ \text{and}\ \mathrm{D}(3; -3; -1).$$
1. Calculation of an angle
a. Calculate the coordinates of the vectors $\overrightarrow{\mathrm{AB}}$ and $\overrightarrow{\mathrm{AC}}$ and deduce that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are not collinear. b. Calculate the lengths AB and AC. c. Using the dot product $\overrightarrow{\mathrm{AB}} \cdot \overrightarrow{\mathrm{AC}}$, determine the value of the cosine of the angle $\widehat{\mathrm{BAC}}$ then give an approximate value of the measure of the angle $\widehat{\mathrm{BAC}}$ to the nearest tenth of a degree.
2. Calculation of an area
a. Determine an equation of the plane $\mathscr{P}$ passing through point C and perpendicular to the line (AB). b. Give a parametric representation of the line (AB). c. Deduce the coordinates of the orthogonal projection E of point C onto the line $(\mathrm{AB})$, that is to say the point of intersection of the line (AB) and the plane $\mathscr{P}$. d. Calculate the area of triangle ABC.
3. Calculation of a volume
a. Let the point $\mathrm{F}(1; -1; 3)$. Show that the points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$ and $\mathrm{F}$ are coplanar. b. Verify that the line (FD) is orthogonal to the plane (ABC). c. Knowing that the volume of a tetrahedron is equal to one third of the area of its base multiplied by its height, calculate the volume of the tetrahedron ABCD.

Exercise 2 Consider the cube ABCDEFGH with side length 1. The space is equipped with the orthonormal coordinate system $(A; \overrightarrow{AB}, \overrightarrow{AD}, \overrightarrow{AE})$.

1. a. Justify that the lines (AH) and (ED) are perpendicular. b. Justify that the line (GH) is orthogonal to the plane (EDH). c. Deduce that the line (ED) is orthogonal to the plane (AGH).
2. Give the coordinates of the vector $\overrightarrow{\mathrm{ED}}$. Deduce from question 1.c. that a Cartesian equation of the plane (AGH) is: $$y - z = 0.$$
3. Let L be the point with coordinates $\left(\frac{2}{3}; 1; 0\right)$. a. Determine a parametric representation of the line (EL). b. Determine the intersection of the line (EL) and the plane (AGH). c. Prove that the orthogonal projection of point L onto the plane (AGH) is the point K with coordinates $\left(\frac{2}{3}; \frac{1}{2}; \frac{1}{2}\right)$. d. Show that the distance from point L to the plane (AGH) is equal to $\frac{\sqrt{2}}{2}$. e. Determine the volume of the tetrahedron LAGH. Recall that the volume $V$ of a tetrahedron is given by the formula: $$V = \frac{1}{3} \times (\text{area of the base}) \times \text{height}.$$

Exercise 2 (7 points) Theme: geometry in space In space with respect to an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider:

• the point A with coordinates $(-1; 1; 3)$,
• the line $\mathscr{D}$ whose parametric representation is: $\left\{\begin{aligned} x &= 1 + 2t \\ y &= 2 - t \\ z &= 2 + 2t \end{aligned} \quad t \in \mathbb{R}\right.$.

It is admitted that point A does not belong to line $\mathscr{D}$.

1. a. Give the coordinates of a direction vector $\vec{u}$ of line $\mathscr{D}$. b. Show that point $B(-1; 3; 0)$ belongs to line $\mathscr{D}$. c. Calculate the dot product $\overrightarrow{AB} \cdot \vec{u}$.
2. We denote by $\mathscr{P}$ the plane passing through point A and perpendicular to line $\mathscr{D}$, and we call H the point of intersection of plane $\mathscr{P}$ and line $\mathscr{D}$. Thus, H is the orthogonal projection of A onto line $\mathscr{D}$. a. Show that plane $\mathscr{P}$ has the Cartesian equation: $2x - y + 2z - 3 = 0$. b. Deduce that point H has coordinates $\left(\frac{7}{9}; \frac{19}{9}; \frac{16}{9}\right)$. c. Calculate the length AH. An exact value will be given.
3. In this question, we propose to find the coordinates of point H, the orthogonal projection of point A onto line $\mathscr{D}$, by another method. We recall that point $B(-1; 3; 0)$ belongs to line $\mathscr{D}$ and that vector $\vec{u}$ is a direction vector of line $\mathscr{D}$. a. Justify that there exists a real number $k$ such that $\overrightarrow{HB} = k\vec{u}$. b. Show that $k = \frac{\overrightarrow{AB} \cdot \vec{u}}{\|\vec{u}\|^2}$. c. Calculate the value of the real number $k$ and find the coordinates of point H.
4. We consider a point C belonging to plane $\mathscr{P}$ such that the volume of tetrahedron ABCH is equal to $\frac{8}{9}$. Calculate the area of triangle ACH. We recall that the volume of a tetrahedron is given by: $V = \frac{1}{3} \times \mathscr{B} \times h$ where $\mathscr{B}$ denotes the area of a base and $h$ the height relative to this base.

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.

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.

A house consists of a rectangular parallelepiped ABCDEFGH topped with a prism EFIHGJ whose base is the triangle EIF isosceles at I.
We have $\mathrm { AB } = 3 , \quad \mathrm { AD } = 2 , \quad \mathrm { AE } = 1$. We define the vectors $\vec { \imath } = \frac { 1 } { 3 } \overrightarrow { \mathrm { AB } } , \vec { \jmath } = \frac { 1 } { 2 } \overrightarrow { \mathrm { AD } } , \vec { k } = \overrightarrow { \mathrm { AE } }$. We thus equip space with the orthonormal coordinate system $( \mathrm { A } ; \vec { \imath } , \vec { \jmath } , \vec { k } )$.

1. Give the coordinates of point G.
2. The vector $\vec { n }$ with coordinates $( 2 ; 0 ; - 3 )$ is a normal vector to the plane (EHI).
   Determine a Cartesian equation of the plane (EHI).
3. Determine the coordinates of point I.
4. Determine a measure to the nearest degree of the angle $\widehat { \mathrm { EIF } }$.
5. In order to connect the house to the electrical network, it is desired to dig a straight trench from an electrical relay located below the house.
   The relay is represented by the point R with coordinates $( 6 ; - 3 ; - 1 )$. The trench is assimilated to a segment of a line $\Delta$ passing through R and directed by the vector $\vec { u }$ with coordinates $( - 3 ; 4 ; 1 )$. It is desired to verify that the trench will reach the house at the level of the edge [BC]. a. Give a parametric representation of the line $\Delta$. b. It is admitted that an equation of the plane (BFG) is $x = 3$.
   Let K be the point of intersection of the line $\Delta$ with the plane (BFG). Determine the coordinates of point K. c. Does the point K indeed belong to the edge $[ \mathrm { BC } ]$?

Exercise 3 — Main topics covered: geometry in space.
A house is modelled by a rectangular parallelepiped ABCDEFGH topped with a pyramid EFGHS. We have $\mathrm{DC} = 6$, $\mathrm{DA} = \mathrm{DH} = 4$. Let the points I, J and K be such that $$\overrightarrow{\mathrm{DI}} = \frac{1}{6}\overrightarrow{\mathrm{DC}}, \quad \overrightarrow{\mathrm{DJ}} = \frac{1}{4}\overrightarrow{\mathrm{DA}}, \quad \overrightarrow{\mathrm{DK}} = \frac{1}{4}\overrightarrow{\mathrm{DH}}.$$ We denote $\vec{\imath} = \overrightarrow{\mathrm{DI}}$, $\vec{\jmath} = \overrightarrow{\mathrm{DJ}}$, $\vec{k} = \overrightarrow{\mathrm{DK}}$. We use the orthonormal coordinate system $(\mathrm{D}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We admit that point S has coordinates $(3; 2; 6)$.

1. Give, without justification, the coordinates of points $\mathrm{B}$, $\mathrm{E}$, $\mathrm{F}$ and G.
   
2. Prove that the volume of the pyramid EFGHS represents one seventh of the total volume of the house. Recall that the volume $V$ of a tetrahedron is given by the formula: $$V = \frac{1}{3} \times (\text{area of the base}) \times \text{height}.$$
   
3. a. Prove that the vector $\vec{n}$ with coordinates $\begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}$ is normal to the plane (EFS). b. Deduce that a Cartesian equation of the plane (EFS) is $y + z - 8 = 0$.
   
4. An antenna is installed on the roof, represented by the segment $[\mathrm{PQ}]$. We have the following data:
   • point P belongs to the plane (EFS);
   • point Q has coordinates $(2; 3; 5{,}5)$;
   • the line (PQ) is directed by the vector $\vec{k}$.
   a. Justify that a parametric representation of the line (PQ) is: $$\left\{\begin{aligned} x &= 2 \\ y &= 3 \\ z &= 5{,}5 + t \end{aligned} \quad (t \in \mathbb{R})\right.$$ b. Deduce the coordinates of point $P$. c. Deduce the length PQ of the antenna.
   
5. A bird flies following a trajectory modelled by the line $\Delta$ whose parametric representation is: $$\left\{\begin{aligned} x &= -4 + 6s \\ y &= 7 - 4s \\ z &= 2 + 4s \end{aligned} \quad (s \in \mathbb{R})\right.$$ Determine the relative position of the lines (PQ) and $\Delta$. Will the bird collide with the antenna represented by the segment $[\mathrm{PQ}]$?