We place ourselves in space with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$.
We consider the point $\mathrm{A}(1; 1; 0)$ and the vector $\vec{u}\begin{pmatrix} 0 \\ 2 \\ -1 \end{pmatrix}$.
We consider the plane $\mathscr{P}$ with equation: $x + 4y + 2z + 1 = 0$.

1. We denote (d) the line passing through A and directed by the vector $\vec{u}$. Determine a parametric representation of (d).
2. Justify that the line (d) and the plane $\mathscr{P}$ intersect at a point B whose coordinates are $(1; -1; 1)$.
3. We consider the point $\mathrm{C}(1; -1; -1)$.
   a. Verify that the points $\mathrm{A}$, $\mathrm{B}$ and C do indeed define a plane.
   b. Show that the vector $\vec{n}\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$ is a normal vector to the plane (ABC).
   c. Determine a Cartesian equation of the plane (ABC).
   
4. a. Justify that the triangle ABC is isosceles at A.
   b. Let H be the midpoint of segment [BC]. Calculate the length AH then the area of triangle ABC.
   
5. Let D be the point with coordinates $(0; -1; 1)$.
   a. Show that the line (BD) is a height of the pyramid ABCD.
   b. Deduce from the previous questions the volume of the pyramid ABCD.

We recall that the volume $V$ of a pyramid is given by: $$V = \frac{1}{3}\mathscr{B} \times h,$$ where $\mathscr{B}$ is the area of a base and $h$ the corresponding height.

Exercise 4

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

1. Prove that the points $\mathrm { A } , \mathrm { B }$ and C define a plane which we will denote $\mathscr { P }$.
2. a. Show that the vector $\vec { n } \left( \begin{array} { c } 13 \\ - 16 \\ - 9 \end{array} \right)$ is normal to the plane $\mathscr { P }$. b. Prove that a Cartesian equation of the plane $\mathscr{P}$ is $13 x - 16 y - 9 z - 17 = 0$.

We denote $\mathscr { D }$ the line passing through the point $\mathrm { F } ( 15 ; - 16 ; - 8 )$ and perpendicular to the plane $\mathscr { P }$.
3. Give a parametric representation of the line $\mathscr { D }$.
4. We call E the point of intersection of the line $\mathscr { D }$ and the plane $\mathscr { P }$. Prove that the point E has coordinates $( 2 ; 0 ; 1 )$.
5. Determine the exact value of the distance from point F to the plane $\mathscr { P }$. 6. Determine the coordinates of the point(s) on the line $\mathscr { D }$ whose distance to the plane $\mathscr { P }$ is equal to half the distance from point F to 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.
Space is referred to an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath}, \vec{k})$.
We consider:

• the points $A(-1; -2; 3)$, $B(1; -2; 7)$ and $C(1; 0; 2)$;
• the line $\Delta$ with parametric representation: $\left\{\begin{array}{l} x = 1 - t \\ y = 2 \\ z = -4 + 3t \end{array}\right.$, where $t \in \mathbb{R}$;
• the plane $\mathscr{P}$ with Cartesian equation: $3x + 2y + z - 4 = 0$;
• the plane $\mathscr{Q}$ with Cartesian equation: $-6x - 4y - 2z + 7 = 0$.

1. Which of the following points belongs to the plane $\mathscr{P}$? a. $R(1; -3; 1)$; b. $S(1; 2; -1)$; c. $T(1; 0; 1)$; d. $U(2; -1; 1)$.
2. Triangle ABC is: a. equilateral; b. right isosceles; c. isosceles non-right; d. right non-isosceles.
3. The line $\Delta$ is: a. orthogonal to the plane $\mathscr{P}$; b. secant to the plane $\mathscr{P}$; c. included in the plane $\mathscr{P}$; d. strictly parallel to the plane $\mathscr{P}$.
4. We are given the dot product $\overrightarrow{BA} \cdot \overrightarrow{BC} = 20$.
   A measure to the nearest degree of the angle $\widehat{ABC}$ is: a. $34°$; b. $120°$; c. $90°$; d. $0°$.
5. The intersection of planes $\mathscr{P}$ and $\mathscr{Q}$ is: a. a plane; b. the empty set; c. a line; d. reduced to a point.

Space is equipped with an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath}, \vec{k})$ in which we consider:

• the points $A(6; -6; 6)$, $B(-6; 0; 6)$ and $C(-2; -2; 11)$.
• the line $(d)$ orthogonal to the two secant lines $(AB)$ and $(BC)$ and passing through point A;
• the line $(d')$ with parametric representation:

$$\left\{\begin{aligned} x &= -6 - 8t \\ y &= 4t, \text{ with } t \in \mathbb{R}. \\ z &= 6 + 5t \end{aligned}\right.$$
This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. A wrong answer, multiple answers or absence of answer to a question neither awards nor deducts points. No justification is required.
Question 1 Among the following vectors, which is a direction vector of the line $(d)$? a. $\overrightarrow{u_1}\left(\begin{array}{c}-6 \\ 3 \\ 0\end{array}\right)$ b. $\overrightarrow{u_2}\left(\begin{array}{l}1 \\ 2 \\ 6\end{array}\right)$ c. $\overrightarrow{u_3}\left(\begin{array}{c}1 \\ 2 \\ 0.2\end{array}\right)$ d. $\overrightarrow{u_4}\left(\begin{array}{l}1 \\ 2 \\ 0\end{array}\right)$
Question 2 Among the following equations, which is a parametric representation of the line (AB)? a. $\left\{\begin{aligned}x &= 2t + 6 \\ y &= -6 \text{ with } t \in \mathbb{R} \\ z &= t + 6\end{aligned}\right.$ b. $\left\{\begin{aligned}x &= 2t - 6 \\ y &= -6 \text{ with } t \in \mathbb{R} \\ z &= -t - 6\end{aligned}\right.$ c. $\left\{\begin{aligned}x &= 2t + 6 \\ y &= -t - 6 \text{ with } t \in \mathbb{R} \\ z &= 6\end{aligned}\right.$ d. $\left\{\begin{aligned}x &= 2t + 6 \\ y &= t - 6 \text{ with } t \in \mathbb{R} \\ z &= 6\end{aligned}\right.$
Question 3
A direction vector of the line $(d')$ is: a. $\overrightarrow{v_1}\left(\begin{array}{c}-6 \\ 0 \\ 6\end{array}\right)$ b. $\overrightarrow{v_2}\left(\begin{array}{c}-14 \\ 4 \\ 11\end{array}\right)$ c. $\overrightarrow{v_3}\left(\begin{array}{c}8 \\ -4 \\ -5\end{array}\right)$ d. $\overrightarrow{v_4}\left(\begin{array}{l}8 \\ 4 \\ 5\end{array}\right)$
Question 4 Which of the following four points belongs to the line $(d')$? a. $M_1(50; -28; -29)$ b. $M_2(-14; -4; 1)$ c. $M_3(2; -4; -1)$ d. $M_4(-3; 0; 3)$
Question 5 The plane with equation $x = 1$ has as normal vector: a. $\overrightarrow{n_1}\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right)$ b. $\overrightarrow{n_2}\left(\begin{array}{l}0 \\ 1 \\ 1\end{array}\right)$ c. $\overrightarrow{n_3}\left(\begin{array}{l}0 \\ 1 \\ 0\end{array}\right)$ d. $\overrightarrow{n_4}\left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right)$

We consider the cube ABCDEFGH below such that $\mathrm { AB } = 1$. We denote M the center of face BCGF and N the center of face EFGH.
We use the orthonormal coordinate system ( D ; $\overrightarrow { \mathrm { DH } } , \overrightarrow { \mathrm { DC } } , \overrightarrow { \mathrm { DA } }$ ).

1. Give without justification the coordinates of points F and C.
2. Calculate the coordinates of points M and N.
3. a. Prove that the vector $\overrightarrow { \mathrm { AG } }$ is normal to the plane (HFC). b. Deduce a Cartesian equation of the plane (HFC).
4. Determine a parametric representation of the line (AG).
5. Prove that the point R with coordinates $\left( \frac { 2 } { 3 } ; \frac { 2 } { 3 } ; \frac { 1 } { 3 } \right)$ is the orthogonal projection of point G onto the plane (HFC).
6. We admit that a parametric representation of the line (FG) is: $$\left\{ \begin{array} { l } x = 1 \\ y = 1 \quad ( t \in \mathbb { R } ) . \\ z = t \end{array} \right.$$ Prove that there exists a unique point K on the line (FG) such that the triangle KMN is right-angled at K.
7. What fraction of the volume of cube ABCDEFGH does the volume of tetrahedron FNKM represent?

Consider a cube ABCDEFGH with side length 1.
The point I is the midpoint of segment [BD]. We define the point L such that $\overrightarrow { \mathrm { IL } } = \frac { 3 } { 4 } \overrightarrow { \mathrm { IG } }$. We use the orthonormal coordinate system ( $A ; \overrightarrow { A B } , \overrightarrow { A D } , \overrightarrow { A E }$ ).

1. a. Specify the coordinates of points $\mathrm { D } , \mathrm { B } , \mathrm { I }$ and G.
   No justification is required. b. Show that point L has coordinates $\left( \frac { 7 } { 8 } ; \frac { 7 } { 8 } ; \frac { 3 } { 4 } \right)$.
2. Verify that a Cartesian equation of plane (BDG) is $x + y - z - 1 = 0$.
3. Consider the line $\Delta$ perpendicular to plane (BDG) passing through L. a. Justify that a parametric representation of line $\Delta$ is: $$\left\{ \begin{aligned} x & = \frac { 7 } { 8 } + t \\ y & = \frac { 7 } { 8 } + t \text { where } t \in \mathbb { R } . \\ z & = \frac { 3 } { 4 } - t \end{aligned} \right.$$ b. Show that lines $\Delta$ and (AE) intersect at point K with coordinates $\left( 0 ; 0 ; \frac { 13 } { 8 } \right)$. c. What does point L represent for point K? Justify your answer.
4. a. Calculate the distance KL. b. We admit that triangle DBG is equilateral. Show that its area equals $\frac { \sqrt { 3 } } { 2 }$. c. Deduce the volume of tetrahedron KDBG.
   We recall that:
   • the volume of a pyramid is given by the formula $V = \frac { 1 } { 3 } \times \mathscr { B } \times h$ where $\mathscr { B }$ is the area of a base and $h$ is the length of the height relative to this base;
   • a tetrahedron is a pyramid with a triangular base.
5. We denote by $a$ a real number belonging to the interval $] 0 ; + \infty \left[ \right.$ and we note $K _ { a }$ the point with coordinates ( $0 ; 0 ; a$ ). a. Express the volume $V _ { a }$ of pyramid $\mathrm { ABCD } K _ { a }$ as a function of $a$. b. We denote $\Delta _ { a }$ the line with parametric representation $$\left\{ \begin{aligned} x & = t ^ { \prime } \\ y & = t ^ { \prime } \\ z & = - t ^ { \prime } + a \end{aligned} \quad \text { where } t ^ { \prime } \in \mathbb { R } . \right.$$ We call $L _ { a }$ the point of intersection of line $\Delta _ { a }$ with plane (BDG). Show that the coordinates of point $L _ { a }$ are $\left( \frac { a + 1 } { 3 } ; \frac { a + 1 } { 3 } ; \frac { 2 a - 1 } { 3 } \right)$. c. Determine, if it exists, a strictly positive real number $a$ such that tetrahedron $\mathrm { GDB } L _ { a }$ and pyramid $\mathrm { ABCD } K _ { a }$ have the same volume.

We consider the cube ABCDEFGH represented below. The points I and J are the midpoints of segments $[\mathrm{AB}]$ and $[\mathrm{CG}]$ respectively. The point N is the midpoint of segment [IJ]. The objective of this exercise is to calculate the volume of the tetrahedron HFIJ. We place ourselves in the orthonormal coordinate system ($A$; $\overrightarrow{AB}, \overrightarrow{AD}, \overrightarrow{AE}$).

1. a. Give the coordinates of points I and J.
   Deduce the coordinates of N. b. Justify that the vectors $\overrightarrow{\mathrm{IJ}}$ and $\overrightarrow{\mathrm{NF}}$ have the respective coordinates: $$\overrightarrow{\mathrm{IJ}} \left(\begin{array}{c} 0.5 \\ 1 \\ 0.5 \end{array}\right) \text{ and } \overrightarrow{\mathrm{NF}} \left(\begin{array}{c} 0.25 \\ -0.5 \\ 0.75 \end{array}\right)$$ c. Prove that the vectors $\overrightarrow{\mathrm{IJ}}$ and $\overrightarrow{\mathrm{NF}}$ are orthogonal.
   We admit that $\mathrm{NF} = \frac{\sqrt{14}}{4}$. d. Deduce that the area of triangle FIJ is equal to $\frac{\sqrt{21}}{8}$.
   
2. We consider the vector $\vec{u}\left(\begin{array}{c} 4 \\ -1 \\ -2 \end{array}\right)$. a. Prove that the vector $\vec{u}$ is normal to the plane (FIJ). b. Deduce that a Cartesian equation of the plane (FIJ) is: $4x - y - 2z - 2 = 0$. c. We denote by $d$ the line perpendicular to the plane (FIJ) passing through point H. Determine a parametric representation of the line $d$. d. Show that the distance from point H to the plane (FIJ) is equal to $\frac{5\sqrt{21}}{21}$. e. We recall that the volume of a pyramid is given by the formula $V = \frac{1}{3} \times \mathscr{B} \times h$ where $\mathscr{B}$ is the area of a base and $h$ is the length of the height relative to this base. Calculate the volume of the tetrahedron HFIJ. Give the answer in the form of an irreducible fraction.

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points. In this exercise, the questions are independent of one another. The four statements are placed in the following situation: In space equipped with an orthonormal reference frame ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ), we consider the points:
$$\mathrm { A } ( 2 ; 1 ; - 1 ) , \quad \mathrm { B } ( - 1 ; 2 ; 1 ) \text { and } \quad \mathrm { C } ( 5 ; 0 ; - 3 ) .$$
We denote $\mathscr { P }$ the plane with Cartesian equation:
$$x + 5 y - 2 z + 3 = 0 .$$
We denote $\mathscr { D }$ the line with parametric representation:
$$\left\{ \begin{array} { r l } x & = - t + 3 \\ y & = t + 2 \\ z & = 2 t + 1 \end{array} , t \in \mathbb { R } . \right.$$
Statement 1: The vector $\vec { n } \left( \begin{array} { l } 1 \\ 0 \\ 2 \end{array} \right)$ is normal to the plane (OAC).
Statement 2: The lines $\mathscr { D }$ and ( AB ) intersect at point C .
Statement 3: The line $\mathscr { D }$ is parallel to the plane $\mathscr { P }$.
Statement 4: The perpendicular bisector plane of segment $[ \mathrm { BC } ]$, denoted $Q$, has Cartesian equation:
$$3 x - y - 2 z - 7 = 0 .$$
Recall that the perpendicular bisector plane of a segment is the plane perpendicular to this segment and passing through its midpoint.

We consider the rectangular prism ABCDEFGH such that $\mathrm{AB} = 3$ and $\mathrm{AD} = \mathrm{AE} = 1$.
We consider the point I on the segment $[\mathrm{AB}]$ such that $\overrightarrow{\mathrm{AB}} = 3\overrightarrow{\mathrm{AI}}$ and we call $M$ the midpoint of the segment [CD]. We place ourselves in the orthonormal coordinate system ($A$; $\overrightarrow{AI}, \overrightarrow{AD}, \overrightarrow{AE}$).

1. Without justification, give the coordinates of the points $\mathrm{F}$, $\mathrm{H}$ and $M$.
2. 1. [a.] Show that the vector $\vec{n}\begin{pmatrix} 2 \\ 6 \\ 3 \end{pmatrix}$ is a normal vector to the plane (HMF).
   2. [b.] Deduce that a Cartesian equation of the plane (HMF) is: $$2x + 6y + 3z - 9 = 0$$
   3. [c.] Is the plane $\mathscr{P}$ whose Cartesian equation is $5x + 15y - 3z + 7 = 0$ parallel to the plane (HMF)? Justify your answer.
3. Determine a parametric representation of the line (DG).
4. We call $N$ the point of intersection of the line (DG) with the plane (HMF). Determine the coordinates of point N.
5. Is the point R with coordinates $\left(3; \frac{1}{4}; \frac{1}{2}\right)$ the orthogonal projection of point G onto the plane (HMF)? Justify your answer.

5 POINTS
In space equipped with an orthonormal reference frame $(O; \vec{\imath}, \vec{\jmath}, \vec{k})$ with unit 1 cm, we consider the points: $A(3; -1; 1)$; $B(4; -1; 0)$; $C(0; 3; 2)$; $D(4; 3; -2)$ and $S(2; 1; 4)$.
In this exercise we wish to show that SABDC is a pyramid with trapezoidal base ABDC and apex $S$, in order to calculate its volume.

1. Show that the points $A$, $B$ and $C$ are not collinear.
2. a. Show that the points $A$, $B$, $C$ and $D$ are coplanar. b. Show that the quadrilateral ABDC is a trapezoid with bases $[AB]$ and $[CD]$.

Recall that a trapezoid is a quadrilateral having two opposite parallel sides called bases.
3. a. Prove that the vector $\vec{n}(2; 1; 2)$ is a normal vector to the plane (ABC). b. Deduce a Cartesian equation of the plane (ABC). c. Determine a parametric representation of the line $\Delta$ passing through point $S$ and orthogonal to the plane (ABC). d. Let I be the point of intersection of the line $\Delta$ and the plane (ABC). Show that point I has coordinates $\left(\frac{2}{3}; \frac{1}{3}; \frac{8}{3}\right)$, then show that $SI = 2$ cm.
4. a. Verify that the orthogonal projection H of point B onto the line (CD) has coordinates $H(3; 3; -1)$ and show that $HB = 3\sqrt{2}$ cm. b. Calculate the exact value of the area of trapezoid ABDC.
Recall that the area of a trapezoid is given by the formula $$\mathscr{A} = \frac{b + B}{2} \times h$$ where $b$ and $B$ are the lengths of the bases of the trapezoid and $h$ is its height.
5. Determine the volume of pyramid SABDC.
Recall that the volume $V$ of a pyramid is given by the formula $$V = \frac{1}{3} \times \text{area of the base} \times \text{height}$$

The cube ABCDEFGH has edge length 1 cm. The point I is the midpoint of segment [AB] and the point J is the midpoint of segment [CG].
We place ourselves in the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$.

1. Give the coordinates of points I and J.
2. Show that the vector $\overrightarrow{\mathrm{EJ}}$ is normal to the plane (FHI).
3. Show that a Cartesian equation of the plane (FHI) is $-2x - 2y + z + 1 = 0$.
4. Determine a parametric representation of the line (EJ).
5. 1. [a.] We denote K the orthogonal projection of point E onto the plane $(\mathrm{FHI})$. Calculate its coordinates.
   2. [b.] Show that the volume of the pyramid EFHI is $\frac{1}{6}\mathrm{~cm}^3$.
   We may use the point L, midpoint of segment $[\mathrm{EF}]$. We admit that this point is the orthogonal projection of point I onto the plane (EFH).
   1. [c.] Deduce from the two previous questions the area of triangle FHI.

Space is equipped with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ). Consider the points $\mathrm { A } ( 5 ; 5 ; 0 ) , \mathrm { B } ( 0 ; 5 ; 0 ) , \mathrm { C } ( 0 ; 0 ; 10 )$ and $\mathrm { D } \left( 0 ; 0 ; - \frac { 5 } { 2 } \right)$.

1. a. Show that $\overrightarrow { n _ { 1 } } \left( \begin{array} { c } 1 \\ - 1 \\ 0 \end{array} \right)$ is a normal vector to the plane (CAD). b. Deduce that the plane (CAD) has the Cartesian equation: $x - y = 0$.
2. Consider the line $\mathscr { D }$ with parametric representation $\left\{ \begin{aligned} x & = \frac { 5 } { 2 } t \\ y & = 5 - \frac { 5 } { 2 } t \text { where } t \in \mathbb { R } \text { . } \\ z & = 0 \end{aligned} \right.$ a. We admit that the line $\mathscr { D }$ and the plane (CAD) intersect at a point H. Justify that the coordinates of H are $\left( \frac { 5 } { 2 } ; \frac { 5 } { 2 } ; 0 \right)$. b. Prove that the point H is the orthogonal projection of B onto the plane (CAD).
3. a. Prove that the triangle ABH is right-angled at H. b. Deduce that the area of triangle ABH is equal to $\frac { 25 } { 4 }$.
4. a. Prove that ( CO ) is the height of the tetrahedron ABCH from C. b. Deduce the volume of the tetrahedron ABCH.

We recall that the volume of a tetrahedron is given by: $V = \frac { 1 } { 3 } \mathscr { B } h$, where $\mathscr { B }$ is the area of a base and h the height relative to this base.
5. We admit that the triangle ABC is right-angled at B. Deduce from the previous questions the distance from point H to the plane (ABC).

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: $\left\{ \begin{array} { l } x = 0 \\ y = 1 + t \\ z = 2 + t \end{array} , t \in \mathbb { R } \right.$.

1. Give a parametric representation of the line $\left( d _ { 1 } \right)$.
2. Prove that the lines $\left( d _ { 1 } \right)$ and $\left( d _ { 2 } \right)$ are non-coplanar.
3. 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: $- 2 x + y + 5 z + 5 = 0$.
4. 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 $\left( - \frac { 2 } { 3 } ; - \frac { 4 } { 3 } ; - 1 \right)$.
5. 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 $\left( d _ { 2 } \right)$.

Space is equipped with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider the three points $\mathrm{A}(3;0;0)$, $\mathrm{B}(0;2;0)$ and $\mathrm{C}(0;0;2)$.
The objective of this exercise is to demonstrate the following property: ``The square of the area of triangle ABC is equal to the sum of the squares of the areas of the three other faces of the tetrahedron OABC''.
Part 1: Distance from point O to the plane (ABC)

1. Demonstrate that the vector $\vec{n}(2;3;3)$ is normal to the plane (ABC).
2. Demonstrate that a Cartesian equation of the plane (ABC) is: $2x + 3y + 3z - 6 = 0$.
3. Give a parametric representation of the line $d$ passing through O and with direction vector $\vec{n}$.
4. We denote H the point of intersection of the line $d$ and the plane (ABC). Determine the coordinates of point H.
5. Deduce that the distance from point O to the plane (ABC) is equal to $\dfrac{3\sqrt{22}}{11}$.

Part 2: Demonstration of the property

1. Demonstrate that the volume of the tetrahedron OABC is equal to 2.
2. Deduce that the area of triangle ABC is equal to $\sqrt{22}$.
3. Demonstrate that for the tetrahedron OABC, ``the square of the area of triangle ABC is equal to the sum of the squares of the areas of the three other faces of the tetrahedron''. Recall that the volume of a tetrahedron is given by $V = \dfrac{1}{3}B \times h$ where $B$ is the area of a base of the tetrahedron and $h$ is the height relative to this base.

In an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$ of space, we consider the plane $(P)$ with equation:
$$(P) : \quad 2x + 2y - 3z + 1 = 0 .$$
We consider the three points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ with coordinates:
$$\mathrm{A}(1;0;1), \quad \mathrm{B}(2;-1;1) \quad \text{and} \quad \mathrm{C}(-4;-6;5).$$
The purpose of this exercise is to study the ratio of areas between a triangle and its orthogonal projection onto a plane.
Part A

1. For each of the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$, verify whether it belongs to the plane $(P)$.
2. Show that the point $\mathrm{C}^{\prime}(0;-2;-1)$ is the orthogonal projection of point $\mathrm{C}$ onto the plane $(P)$.
3. Determine a parametric representation of the line (AB).
4. We admit the existence of a unique point H satisfying the two conditions $$\left\{ \begin{array}{l} \mathrm{H} \in (\mathrm{AB}) \\ (\mathrm{AB}) \text{ and } (\mathrm{HC}) \text{ are orthogonal.} \end{array} \right.$$ Determine the coordinates of point H.

Part B
We admit that the coordinates of the vector $\overrightarrow{\mathrm{HC}}$ are: $\overrightarrow{\mathrm{HC}} \left( \begin{array}{c} -\frac{11}{2} \\ -\frac{11}{2} \\ 4 \end{array} \right)$.

1. Calculate the exact value of $\| \overrightarrow{\mathrm{HC}} \|$.
2. Let $S$ be the area of triangle ABC. Determine the exact value of $S$.

Part C
We admit that $\mathrm{HC}^{\prime} = \sqrt{\frac{17}{2}}$.

1. Let $\alpha = \widehat{\mathrm{CHC}^{\prime}}$. Determine the value of $\cos(\alpha)$.
2. a. Show that the lines $(\mathrm{C}^{\prime}\mathrm{H})$ and (AB) are perpendicular. b. Calculate $S^{\prime}$ the area of triangle $\mathrm{ABC}^{\prime}$, give the exact value. c. Give a relationship between $S$, $S^{\prime}$ and $\cos(\alpha)$.

Space is equipped with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. Consider:

• the points $\mathrm{A}(-2; 0; 2)$, $\mathrm{B}(-1; 3; 0)$, $\mathrm{C}(1; -1; 2)$ and $\mathrm{D}(0; 0; 3)$.
• the line $\mathscr{D}_1$ whose parametric representation is $\left\{ \begin{aligned} x &= t \\ y &= 3t \\ z &= 3 + 5t \end{aligned} \right.$ with $t \in \mathbb{R}$.
• the line $\mathscr{D}_2$ whose parametric representation is $\left\{ \begin{aligned} x &= 1 + 3s \\ y &= -1 - 5s \\ z &= 2 - 6s \end{aligned} \right.$ with $s \in \mathbb{R}$.

1. Prove that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are not collinear.
2. 1. [a.] Prove that the vector $\vec{n}\begin{pmatrix} 1 \\ 3 \\ 5 \end{pmatrix}$ is orthogonal to the plane (ABC).
   2. [b.] Justify that a Cartesian equation of the plane (ABC) is: $$x + 3y + 5z - 8 = 0$$
   3. [c.] Deduce that the points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$ and $\mathrm{D}$ are not coplanar.
3. 1. [a.] Justify that the line $\mathscr{D}_1$ is the altitude of the tetrahedron ABCD from D. It is admitted that the line $\mathscr{D}_2$ is the altitude of the tetrahedron ABCD from C.
   2. [b.] Prove that the lines $\mathscr{D}_1$ and $\mathscr{D}_2$ are secant and determine the coordinates of their point of intersection.
4. 1. [a.] Determine the coordinates of the orthogonal projection H of point D onto the plane (ABC).
   2. [b.] Calculate the distance from point D to the plane (ABC). Round the result to the nearest hundredth.

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.
In space with an orthonormal coordinate system, consider the following points: $$\mathrm{A}(2;0;0), \quad \mathrm{B}(0;4;3), \quad \mathrm{C}(4;4;1), \quad \mathrm{D}(0;0;4) \text{ and } \mathrm{H}(-1;1;2)$$
Statement 1: the points A, C and D define a plane $\mathscr{P}$ with equation $8x - 5y + 4z - 16 = 0$. Statement 2: the points A, B, C and D are coplanar. Statement 3: the lines $(\mathrm{AC})$ and $(\mathrm{BH})$ are secant. It is admitted that the plane (ABC) has the Cartesian equation $x - y + 2z - 2 = 0$. Statement 4: the point H is the orthogonal projection of point D onto the plane (ABC).

Consider a cube ABCDEFGH and the space is referred to the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$. For any real $m$ belonging to the interval $[0; 1]$, we consider the points $K$ and $L$ with coordinates: $$K(m; 0; 0) \text{ and } L(1-m; 1; 1).$$

1. Give the coordinates of points E and C in this coordinate system.
2. In this question, $m = 0$. Thus, the point $\mathrm{L}(1; 1; 1)$ coincides with point G, the point $\mathrm{K}(0; 0; 0)$ coincides with point A and the plane (LEK) is therefore the plane (GEA). a. Justify that the vector $\overrightarrow{\mathrm{DB}} \left(\begin{array}{c} 1 \\ -1 \\ 0 \end{array}\right)$ is normal to the plane (GEA). b. Determine a Cartesian equation of the plane (GEA).
3. In this question, $m$ is any real number in the interval $[0; 1]$. a. Prove that $\mathrm{CKEL}$ is a parallelogram. b. Justify that $\overrightarrow{KC} \cdot \overrightarrow{KE} = m(m-1)$. c. Prove that $\mathrm{CKEL}$ is a rectangle if, and only if, $m = 0$ or $m = 1$.
4. In this question, $m = \frac{1}{2}$. Thus, L has coordinates $\left(\frac{1}{2}; 1; 1\right)$ and K has coordinates $\left(\frac{1}{2}; 0; 0\right)$. a. Prove that the parallelogram CKEL is then a rhombus. b. Using question 3.b., determine an approximate value to the nearest degree of the measure of the angle $\widehat{\mathrm{CKE}}$.

A municipality decides to replace the traditional July 14 fireworks with a luminous drone show. For drone piloting, space is equipped with an orthonormal reference frame $(\mathrm{O};\vec{\imath},\vec{\jmath},\vec{k})$ whose unit is one hundred meters.
The position of each drone is modeled by a point and each drone is sent from a starting point D with coordinates $(2;5;1)$. It is desired to form figures with drones by positioning them in the same plane $\mathscr{P}$. Three drones are positioned at points $\mathrm{A}(-1;-1;17)$, $\mathrm{B}(4;-2;4)$ and $\mathrm{C}(1;-3;7)$.

1. Justify that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are not collinear.
   In the following, we denote by $\mathscr{P}$ the plane (ABC) and we consider the vector $\vec{n}\begin{pmatrix}2\\-3\\1\end{pmatrix}$.
   
2. a. Justify that $\vec{n}$ is normal to the plane (ABC). b. Prove that a Cartesian equation of the plane $\mathscr{P}$ is $2x - 3y + z - 18 = 0$.
   
3. The drone pilot decides to send a fourth drone taking as trajectory the line $d$ whose parametric representation is given by $$d : \left\{\begin{array}{rl} x &= 3t + 2 \\ y &= t + 5 \\ z &= 4t + 1 \end{array},\text{ with } t \in \mathbb{R}.\right.$$ a. Determine a direction vector of the line $d$. b. So that this new drone is also placed in the plane $\mathscr{P}$, determine by calculation the coordinates of point E, the intersection of the line $d$ with the plane $\mathscr{P}$.
   
4. The drone pilot decides to send a fifth drone along the line $\Delta$ which passes through point $\mathrm{D}$ and which is perpendicular to the plane $\mathscr{P}$. This fifth drone is also placed in the plane $\mathscr{P}$, at the intersection between the line $\Delta$ and the plane $\mathscr{P}$. We admit that the point $\mathrm{F}(6;-1;3)$ corresponds to this location. Prove that the distance between the starting point D and the plane $\mathscr{P}$ equals $2\sqrt{14}$ hundreds of meters.
   
5. The show organizer asks the pilot to send a new drone in the plane (no matter its position in the plane), always starting from point D. Knowing that there are 40 seconds left before the start of the show and that the drone flies in a straight trajectory at $18.6\,\mathrm{m.s}^{-1}$, can the new drone arrive on time?

Space is referred to an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ). We consider:

• $\alpha$ any real number;
• the points $\mathrm { A } ( 1 ; 1 ; 0 ) , \mathrm { B } ( 2 ; 1 ; 0 )$ and $\mathrm { C } ( \alpha ; 3 ; \alpha )$;
• (d) the line with parametric representation:

$$\left\{ \begin{array} { l } x = 1 + t \\ y = 2 t , \quad t \in \mathbb { R } \\ z = - t \end{array} \right.$$ For each of the following statements, specify whether it is true or false, then justify the answer given. An answer without justification will not be taken into account. Statement 1: For all values of $\alpha$, the points $A , B$ and $C$ define a plane and a normal vector to this plane is $\vec { J } \left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)$. Statement 2: There exists exactly one value of $\alpha$ such that the lines ( $A C$ ) and (d) are parallel. Statement 3: A measure of the angle $\widehat { \mathrm { OAB } }$ is $135 ^ { \circ }$. Statement 4: The orthogonal projection of point $A$ onto the line (d) is the point $\mathrm { H } ( 1 ; 2 ; 2 )$. Statement 5: The sphere with center $O$ and radius 1 intersects the line $( d )$ at two distinct points. Recall that the sphere with center $\Omega$ and radius $r$ is the set of points in space at distance $r$ from $\Omega$.

For each of the four following statements, indicate whether it is true or false, by justifying the answer. An unjustified answer is not taken into account. An absence of answer is not penalised.
Consider a cube ABCDEFGH with edge length 1 and the point I defined by $\overrightarrow { \mathrm { FI } } = \frac { 1 } { 3 } \overrightarrow { \mathrm { FB } }$. One may place oneself in the orthonormal coordinate system of space $( \mathrm { A } ; \overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { AE } } )$.

1. Consider the triangle HAC.

Statement 1: The triangle HAC is a right-angled triangle.
2. Consider the lines (HF) and (DI).
Statement 2: The lines (HF) and (DI) are secant.
3. Consider a real number $\alpha$ belonging to the interval $] 0 ; \pi [$.
Consider the vector $\vec { u }$ with coordinates $\left( \begin{array} { c } \sin ( \alpha ) \\ \sin ( \pi - \alpha ) \\ \sin ( - \alpha ) \end{array} \right)$. Statement 3: The vector $\vec { u }$ is a normal vector to the plane (FAC).
4. The cube ABCDEFGH has 8 vertices. We are interested in the number $N$ of segments that can be constructed by connecting 2 distinct vertices of the cube. Statement 4: $N = \frac { 8 ^ { 2 } } { 2 }$.

This exercise is a multiple choice questionnaire. For each question, only one of the three propositions is correct.
In all the following questions, space is referred to an orthonormal coordinate system.

1. Consider the line $\Delta_1$ with parametric representation $\left\{ \begin{aligned} x &= 1 - 3t \\ y &= 4 + 2t \\ z &= t \end{aligned} \right.$, where $t \in \mathbb{R}$ as well as the line $\Delta_2$ with parametric representation $\left\{ \begin{aligned} x &= -4 + s \\ y &= 2 + 2s \\ z &= -1 + s \end{aligned} \right.$, where $s \in \mathbb{R}$. a. The lines $\Delta_1$ and $\Delta_2$ are parallel. b. The lines $\Delta_1$ and $\Delta_2$ are orthogonal. c. The lines $\Delta_1$ and $\Delta_2$ are secant.
2. Consider the line $d$ with parametric representation $\left\{ \begin{aligned} x &= 1 + t \\ y &= 3 - t \\ z &= 1 + 2t \end{aligned} \right.$, where $t \in \mathbb{R}$, and the plane $P$ with Cartesian equation: $4x + 2y - z + 3 = 0$. a. The line $d$ is contained in the plane $P$. b. The line $d$ is strictly parallel to the plane $P$. c. The line $d$ is secant to the plane $P$.
3. Consider the points $\mathrm{A}(3;2;1)$, $\mathrm{B}(7;3;1)$, $\mathrm{C}(-1;4;5)$ and $\mathrm{D}(-3;3;5)$. a. The points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$ and D are not coplanar. b. The points $\mathrm{A}$, $\mathrm{B}$ and C are collinear. c. $\overrightarrow{\mathrm{AB}}$ and $\overrightarrow{\mathrm{CD}}$ are collinear.
4. Consider the planes $Q$ and $Q'$ with respective Cartesian equations $3x - 2y + z + 1 = 0$ and $4x + y - z + 3 = 0$. a. The point $\mathrm{R}(1;1;-2)$ belongs to both planes. b. The two planes are orthogonal. c. The two planes are secant with intersection the line with parametric representation $$\left\{ \begin{aligned} x &= t \\ y &= 7t + 4, \text{ where } t \in \mathbb{R}. \\ z &= 11t + 7 \end{aligned} \right.$$

``In a non-equilateral triangle, the Euler line is the line that passes through the following three points:

• the center of the circumscribed circle of this triangle (circle passing through the three vertices of this triangle).
• the centroid of this triangle located at the intersection of the medians of this triangle.
• the orthocenter of this triangle located at the intersection of the altitudes of this triangle''.

The purpose of the exercise is to study an example of an Euler line. We consider a cube ABCDEFGH with side length one unit. The space is equipped with the orthonormal coordinate system $( \mathrm { A } ; \overrightarrow { \mathrm { AB } } ; \overrightarrow { \mathrm { AD } } ; \overrightarrow { \mathrm { AE } } )$. We denote I the midpoint of segment [AB] and J the midpoint of segment [BG].

1. Give without justification the coordinates of points A, B, G, I and J.
2. a. Determine a parametric representation of the line (AJ). b. Show that a parametric representation of the line (IG) is: $$\left\{ \begin{aligned} x & = \frac { 1 } { 2 } + \frac { 1 } { 2 } t \\ y & = t \\ z & = t \end{aligned} \text { with } t \in \mathbb { R } . \right.$$ c. Prove that the lines (AJ) and (IG) intersect at a point S with coordinates $S \left( \frac { 2 } { 3 } ; \frac { 1 } { 3 } ; \frac { 1 } { 3 } \right)$.
3. a. Show that the vector $\vec { n } ( 0 ; - 1 ; 1 )$ is normal to the plane (ABG). b. Deduce a Cartesian equation of the plane (ABG). c. We admit that a parametric representation of the line (d) with direction vector $\vec { n }$ and passing through the point K with coordinates $\left( \frac { 1 } { 2 } ; 0 ; 1 \right)$ is: $$\left\{ \begin{array} { l } x = \frac { 1 } { 2 } \\ y = - t \quad \text { with } t \in \mathbb { R } . \\ z = 1 + t \end{array} \right.$$ Show that this line (d) intersects the plane (ABG) at a point L with coordinates $L \left( \frac { 1 } { 2 } ; \frac { 1 } { 2 } ; \frac { 1 } { 2 } \right)$. d. Show that the point L is equidistant from the points $\mathrm { A } , \mathrm { B }$ and G.
4. Show that the triangle ABG is right-angled at B.
5. a. Identify the center of the circumscribed circle, the centroid and the orthocenter of triangle ABG (no justification is expected). b. Verify by calculation that these three points are indeed collinear.

In space with respect to an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider the points:
$$A(4; -4; 4), \quad B(5; -3; 2), \quad C(6; -2; 3), \quad D(5; 1; 1)$$

1. Prove that triangle ABC is right-angled at $B$.
2. Justify that a Cartesian equation of the plane (ABC) is: $$x - y - 8 = 0.$$
3. We denote $d$ the line passing through point $D$ and perpendicular to the plane (ABC). a. Determine a parametric representation of the line $d$. b. We denote H the orthogonal projection of point $D$ onto the plane $(ABC)$. Determine the coordinates of point H. c. Show that $DH = 2\sqrt{2}$.
4. a. Show that the volume of the pyramid ABCD is equal to 2. We recall that the volume V of a pyramid is calculated using the formula: $$V = \frac{1}{3} \times \mathscr{B} \times h$$ where $\mathscr{B}$ is the area of a base of the pyramid and $h$ is the corresponding height. b. We admit that the area of triangle BCD is equal to $\frac{\sqrt{42}}{2}$. Deduce the exact value of the distance from point A to the plane (BCD).

This exercise is a multiple choice questionnaire. For each question, only one of the four proposed answers is correct. No justification is required. A wrong answer, multiple answers, or the absence of an answer earns neither points nor deducts points.
Throughout the exercise, we consider that space is equipped with an orthonormal reference frame $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider:

• the points $\mathrm{A}(-3; 1; 4)$ and $\mathrm{B}(1; 5; 2)$
• the plane $\mathscr{P}$ with Cartesian equation $4x + 4y - 2z + 3 = 0$
• the line $(d)$ with parametric representation $\left\{\begin{aligned} x &= -6 + 3t \\ y &= 1 \\ z &= 9 - 5t \end{aligned}\right.$, where $t \in \mathbb{R}$.

1. The lines $(\mathrm{AB})$ and $(d)$ are: a. secant and non-perpendicular. b. perpendicular. c. non-coplanar. d. parallel.
2. The line $(\mathrm{AB})$ is: a. included in the plane $\mathscr{P}$. b. strictly parallel to the plane $\mathscr{P}$. c. secant and non-orthogonal to the plane $\mathscr{P}$. d. orthogonal to the plane $\mathscr{P}$.
3. We consider the plane $\mathscr{P}'$ with Cartesian equation $2x + y + 6z + 5 = 0$. The planes $\mathscr{P}$ and $\mathscr{P}'$ are: a. secant and non-perpendicular. b. perpendicular. c. identical. d. strictly parallel.
4. We consider the point $\mathrm{C}(0; 1; -1)$. The value of the angle $\widehat{\mathrm{BAC}}$ rounded to the nearest degree is: a. $90^\circ$ b. $51^\circ$ c. $39^\circ$ d. $0^\circ$