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.

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.

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.

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 $$V = \frac{1}{3} \times \text{area of the base} \times \text{height}$$

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.

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.

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.

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).

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?

``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).

Exercise 4

We place ourselves in an orthonormal frame $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$ of space. We consider the points $\mathrm{A}(1; 0; 3)$, $\mathrm{B}(-2; 1; 2)$ and $\mathrm{C}(0; 3; 2)$.

1. a. Show that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are not collinear. b. Let $\vec{n}$ be the vector with coordinates $\left(\begin{array}{c} -1 \\ 1 \\ 4 \end{array}\right)$. Verify that the vector $\vec{n}$ is orthogonal to the plane (ABC). c. Deduce that the plane $(\mathrm{ABC})$ has for Cartesian equation $-x + y + 4z - 11 = 0$.

We consider the plane $\mathscr{P}$ with Cartesian equation $3x - 3y + 2z - 9 = 0$ and the plane $\mathscr{P}'$ with Cartesian equation $x - y - z + 2 = 0$.

1. a. Prove that the planes $\mathscr{P}$ and $\mathscr{P}'$ are secant. We denote by (d) their line of intersection. b. Determine whether the planes $\mathscr{P}$ and $\mathscr{P}'$ are perpendicular.
2. Show that the line (d) is directed by the vector $\vec{u}\left(\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right)$.
3. Show that the point $\mathrm{M}(2; 1; 3)$ belongs to the planes $\mathscr{P}$ and $\mathscr{P}'$. Deduce a parametric representation of the line (d).
4. Show that the line (d) is also included in the plane (ABC). What can we say about the three planes (ABC), $\mathscr{P}$ and $\mathscr{P}'$?

Exercise 2
Consider the cube ABCDEFGH. We place the point M such that $\overrightarrow{\mathrm{BM}} = \overrightarrow{\mathrm{AB}}$.
Part A

1. Show that the lines (FG) and (FM) are perpendicular.
2. Show that the points A, M, G and H are coplanar.

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

1. Determine the coordinates of the vectors $\overrightarrow{\mathrm{GM}}$ and $\overrightarrow{\mathrm{AH}}$ and show that they are not collinear.
2. 1. [a.] Justify that a parametric representation of the line (GM) is: $$\left\{\begin{aligned} x &= 1+t \\ y &= 1-t \quad \text{with } t \in \mathbb{R}. \\ z &= 1-t \end{aligned}\right.$$
   2. [b.] We admit that a parametric representation of the line (AH) is: $$\left\{\begin{aligned} x &= 0 \\ y &= k \\ z &= k \end{aligned}\right. \text{ with } k \in \mathbb{R}.$$ Show that the intersection point of (GM) and (AH), which we will call N, has coordinates $(0;2;2)$.
3. 1. [a.] Show that the triangle AMN is a right-angled triangle at A.
   2. [b.] Calculate the area of this triangle.
4. Let J be the centre of the face BCGF.
   1. [a.] Determine the coordinates of point J.
   2. [b.] Show that the vector $\overrightarrow{\mathrm{FJ}}$ is a normal vector to the plane (AMN).
   3. [c.] Show that J belongs to the plane (AMN). Deduce that it is the orthogonal projection of point F onto the plane (AMN).
5. We recall that the volume $V$ of a tetrahedron or a pyramid is given by the formula: $$V = \frac{1}{3} \times \mathscr{B} \times h$$ $\mathscr{B}$ being the area of a base and $h$ the height relative to this base. Show that the volume of the tetrahedron AMNF is twice the volume of the pyramid BCGFM.

Space is referred to an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider the points $$\mathrm{A}(4; -1; 3), \quad \mathrm{B}(-1; 1; -2), \quad \mathrm{C}(0; 4; 5) \text{ and } \mathrm{D}(-3; -4; 6).$$

1. a. Verify that the points $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are not collinear.
   We admit that a Cartesian equation of the plane (ABC) is: $29x + 30y - 17z = 35$. b. Are the points A, B, C, D coplanar? Justify.
   
2. Let $P _ { 1 }$ be the perpendicular bisector plane of the segment $[\mathrm{AB}]$. a. Determine the coordinates of the midpoint of the segment $[\mathrm{AB}]$. b. Deduce that a Cartesian equation of $P _ { 1 }$ is: $5x - 2y + 5z = 10$.
   
3. We denote by $P _ { 2 }$ the perpendicular bisector plane of the segment $[\mathrm{CD}]$. a. Let M be a point of the plane $P _ { 2 }$ with coordinates $(x; y; z)$.
   Express $\mathrm{MC}^{2}$ and $\mathrm{MD}^{2}$ as functions of the coordinates of M. Deduce that a Cartesian equation of the plane $P _ { 2 }$ is: $-3x - 8y + z = 10$. b. Justify that the planes $P _ { 1 }$ and $P _ { 2 }$ are secant.
   
4. Let $\Delta$ be the line with a parametric representation: $$\left\{ \begin{array} { r l r l } x & = & -2 - 1.9t \\ y & = & t & \text{ where } t \in \mathbb{R} \\ z & = & 4 + 2.3t \end{array} \right.$$ Demonstrate that $\Delta$ is the line of intersection of $P _ { 1 }$ and $P _ { 2 }$.
   We denote by $P _ { 3 }$ the perpendicular bisector plane of the segment $[\mathrm{AC}]$. We admit that a Cartesian equation of the plane $P _ { 3 }$ is: $8x - 10y - 4z = -15$.
   
5. Demonstrate that the line $\Delta$ and the plane $P _ { 3 }$ are secant.
   
6. Justify that the point of intersection between $\Delta$ and $P _ { 3 }$ is the point H.

Exercise 2 (5 points)
The space is equipped with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider:

• the points $\mathrm{A}(-1; 2; 1)$, $\mathrm{B}(1; -1; 2)$ and $\mathrm{C}(1; 1; 1)$;
• the line $d$ whose parametric representation is given by: $$d : \left\{ \begin{aligned} x &= \frac{3}{2} + 2t \\ y &= 2 + t \\ z &= 3 - t \end{aligned} \quad \text{with } t \in \mathbb{R}; \right.$$
• the line $d'$ whose parametric representation is given by: $$d' : \left\{ \begin{aligned} x &= s \\ y &= \frac{3}{2} + s \\ z &= 3 - 2s \end{aligned} \quad \text{with } s \in \mathbb{R}. \right.$$

Part A

1. Show that the lines $d$ and $d'$ intersect at the point $\mathrm{S}\left(-\frac{1}{2}; 1; 4\right)$.
2. 1. [a.] Show that the vector $\vec{n}\begin{pmatrix}1\\2\\4\end{pmatrix}$ is a normal vector to the plane (ABC).
   2. [b.] Deduce that a Cartesian equation of the plane (ABC) is: $$x + 2y + 4z - 7 = 0$$
3. Prove that the points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$ and S are not coplanar.
4. 1. [a.] Prove that the point $\mathrm{H}(-1; 0; 2)$ is the orthogonal projection of S onto the plane (ABC).
   2. [b.] Deduce that there is no point $M$ in the plane (ABC) such that $\mathrm{S}M < \frac{\sqrt{21}}{2}$.

Part B
We consider a point $M$ belonging to the segment [CS]. We thus have $\overrightarrow{\mathrm{CM}} = k\overrightarrow{\mathrm{CS}}$ with $k$ a real number in the interval $[0; 1]$.

1. Determine the coordinates of point $M$ as a function of $k$.
2. Does there exist a point $M$ on the segment [CS] such that the triangle $(MAB)$ is right-angled at $M$?

Two aircraft are approaching an airport. We equip space with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$ whose origin O is the base of the control tower, and the ground is the plane $P_0$ with equation $z = 0$. The unit of the axes corresponds to 1 km. We model the aircraft as points.
Aircraft Alpha transmits to the tower its position at $\mathrm{A}(-7; 1; 7)$ and its trajectory is directed by the vector $\vec{u}\begin{pmatrix} 2 \\ -1 \\ -3 \end{pmatrix}$.
Aircraft Beta transmits a trajectory defined by the line $d_{\mathrm{B}}$ passing through point B with a parametric representation: $$\left\{\begin{aligned} x &= -11 + 5t \\ y &= -5 + t \\ z &= 11 - 4t \end{aligned}\right. \text{ where } t \text{ describes } \mathbb{R}.$$

1. If it does not deviate from its trajectory, determine the coordinates of point S where aircraft Beta will touch the ground.
2. a. Determine a parametric representation of the line $d_{\mathrm{A}}$ characterizing the trajectory of aircraft Alpha. b. Can the two aircraft collide?
3. a. Prove that aircraft Alpha passes through position $\mathrm{E}(-3; -1; 1)$. b. Justify that a Cartesian equation of the plane $P_{\mathrm{E}}$ passing through E and perpendicular to the line $d_{\mathrm{A}}$ is: $$2x - y - 3z + 8 = 0.$$ c. Verify that the point $\mathrm{F}(-1; -3; 3)$ is the intersection point of the plane $P_{\mathrm{E}}$ and the line $d_{\mathrm{B}}$. d. Calculate the exact value of the distance EF, then verify that this corresponds to a distance of 3464 m, to the nearest 1 m.
4. Air traffic regulations stipulate that two aircraft on approach must be at least 3 nautical miles apart at all times (1 nautical mile equals 1852 m). If aircraft Alpha and Beta are respectively at E and F at the same instant, is their safety distance respected?

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

1. Show that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ determine a plane.
2. Show that the triangle ABC is right-angled at A.
3. Let $\Delta$ be the line passing through point D and with direction vector $\vec{u}\begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}$. a. Prove that the line $\Delta$ is orthogonal to the plane (ABC). b. Justify that the plane (ABC) admits the Cartesian equation: $$2x - y + z + 1 = 0$$ c. Determine a parametric representation of the line $\Delta$.
4. We call H the point with coordinates $\left(-\dfrac{2}{3}; \dfrac{4}{3}; \dfrac{5}{3}\right)$. Verify that $H$ is the orthogonal projection of point $D$ onto the plane (ABC).
5. We 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 its height relative to this base. a. Show that $\mathrm{DH} = \dfrac{2\sqrt{6}}{3}$. b. Deduce the volume of the tetrahedron ABCD.
6. We consider the line $d$ with parametric representation: $$\left\{\begin{aligned} x &= 1 - 2k \\ y &= -3k \\ z &= 1 + k \end{aligned}\right. \text{ where } k \text{ describes } \mathbb{R}.$$ Are the line $d$ and the plane (ABC) secant or parallel?

As shown in the figure, in quadrilateral $ABCD$, $AB \parallel CD$, $\angle DAB = 90°$, $F$ is the midpoint of $CD$, point $E$ is on $AB$, $EF \parallel AD$, $AB = 3AD$, $CD = 2AD$. Fold quadrilateral $EFDA$ along $EF$ to quadrilateral $EFD'A'$ such that the dihedral angle between plane $EFD'A'$ and plane $EFCB$ is $60°$.
(1) Prove: $A'B \parallel$ plane $CD'F$;
(2) Find the sine of the dihedral angle between plane $BCD'$ and plane $EFD'A'$.

(15 points) In the quadrangular pyramid $P - ABCD$, $PA \perp$ plane $ABCD$, $BC \parallel AD$, $AB \perp AD$.
(1) Prove that plane $PAB \perp$ plane $PAD$.
(2) If $PA = AB = \sqrt{2}$, $AD = \sqrt{3} + 1$, $BC = 2$, and $P, B, C, D$ lie on the same sphere with center $O$.
(i) Prove that $O$ lies on plane $ABCD$.
(ii) Find the cosine of the angle between line $AC$ and line $PO$.

Let $L _ { 1 }$ be the line of intersection of the planes given by the equations
$$2 x + 3 y + z = 4 \text { and } x + 2 y + z = 5 .$$
Let $L _ { 2 }$ be the line passing through the point $P ( 2 , - 1,3 )$ and parallel to $L _ { 1 }$. Let $M$ denote the plane given by the equation
$$2 x + y - 2 z = 6$$
Suppose that the line $L _ { 2 }$ meets the plane $M$ at the point $Q$. Let $R$ be the foot of the perpendicular drawn from $P$ to the plane $M$.
Then which of the following statements is (are) TRUE?

(A)	The length of the line segment $PQ$ is $9 \sqrt { 3 }$
(B)	The length of the line segment $QR$ is 15
(C)	The area of $\triangle PQR$ is $\frac { 3 } { 2 } \sqrt { 234 }$
(D)	The acute angle between the line segments $PQ$ and $PR$ is $\cos ^ { - 1 } \left( \frac { 1 } { 2 \sqrt { 3 } } \right)$

In coordinate space, let $O$ be the origin and $E$ be the plane $x - z = 4$.
Continuing from question 19, it is known that point $P$ is on plane $E$ and $b = 0$. Find the maximum possible range of $c$ and the minimum possible length of line segment $\overline{OP}$. (Non-multiple choice question, 8 points)