Exercise 4 — 7 points

Topics: Geometry in space In space with respect to an orthonormal coordinate system $(\mathrm { O } , \vec { \imath } , \vec { \jmath } , \vec { k })$, we consider:

• the line $\mathscr { D }$ passing through the point $\mathrm { A } ( 2 ; 4 ; 0 )$ and whose direction vector is $\vec { u } \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right)$;
• the line $\mathscr { D } ^ { \prime }$ whose parametric representation is: $\left\{ \begin{array} { r l } x & = 3 \\ y & = 3 + t \\ z & = 3 + t \end{array} , t \in \mathbb { R } \right.$.

1. a. Give the coordinates of a direction vector $\overrightarrow { u ^ { \prime } }$ of the line $\mathscr { D } ^ { \prime }$. b. Show that the lines $\mathscr { D }$ and $\mathscr { D } ^ { \prime }$ are not parallel. c. Determine a parametric representation of the line $\mathscr { D }$.

We admit in the rest of this exercise that there exists a unique line $\Delta$ perpendicular to the lines $\mathscr { D }$ and $\mathscr { D } ^ { \prime }$. This line $\Delta$ intersects each of the lines $\mathscr { D }$ and $\mathscr { D } ^ { \prime }$. We will call M the intersection point of $\Delta$ and $\mathscr { D }$, and $\mathrm { M } ^ { \prime }$ the intersection point of $\Delta$ and $\mathscr { D } ^ { \prime }$. We propose to determine the distance $\mathrm { MM } ^ { \prime }$ called the ``distance between the lines $\mathscr { D }$ and $\mathscr { D } ^ { \prime }$''.

1. Show that the vector $\vec { v } \left( \begin{array} { c } 2 \\ - 1 \\ 1 \end{array} \right)$ is a direction vector of the line $\Delta$.
2. We denote by $\mathscr { P }$ the plane containing the lines $\mathscr { D }$ and $\Delta$, that is, the plane passing through point A and with direction vectors $\vec { u }$ and $\vec { v }$. a. Show that the vector $\vec { n } \left( \begin{array} { c } 2 \\ - 1 \\ - 5 \end{array} \right)$ is a normal vector to the plane $\mathscr { P }$. b. Deduce that an equation of the plane $\mathscr { P }$ is: $2 x - y - 5 z = 0$. c. We recall that $\mathrm { M } ^ { \prime }$ is the intersection point of the lines $\Delta$ and $\mathscr { D } ^ { \prime }$. Justify that $\mathrm { M } ^ { \prime }$ is also the intersection point of $\mathscr { D } ^ { \prime }$ and the plane $\mathscr { P }$. Deduce that the coordinates of point $\mathrm { M } ^ { \prime }$ are $( 3 ; 1 ; 1 )$.
3. a. Determine a parametric representation of the line $\Delta$. b. Justify that point M has coordinates $( 1 ; 2 ; 0 )$. c. Calculate the distance $\mathrm { MM } ^ { \prime }$.
4. We consider the line $d$ with parametric representation $\left\{ \begin{aligned} x & = 5 t \\ y & = 2 + 5 t \\ z & = 1 + t \end{aligned} \right.$ with $t \in \mathbb { R }$. a. Show that the line $d$ is parallel to the plane $\mathscr { P }$. b. We denote by $\ell$ the distance from a point N of the line $d$ to the plane $\mathscr { P }$. Express the volume of the tetrahedron $\mathrm { ANMM } ^ { \prime }$ as a function of $\ell$. We recall that the volume of a tetrahedron is given by: $V = \frac { 1 } { 3 } \times B \times h$ where $B$ denotes the area of a base and $h$ the height relative to this base. c. Justify that, if $\mathrm { N } _ { 1 }$ and $\mathrm { N } _ { 2 }$ are any two points of the line $d$, the tetrahedra $A N _ { 1 } M M ^ { \prime }$ and $A N _ { 2 } M M ^ { \prime }$ have the same volume.

Exercise 4 (7 points) -- Geometry in the plane and in space
Consider the cube ABCDEFGH. Let I be the midpoint of segment [EH] and consider the triangle CFI. The space is equipped with the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$ and we admit that point I has coordinates $\left(0; \frac{1}{2}; 1\right)$ in this coordinate system.

1. a. Give without justification the coordinates of points C, F and G. b. Prove that the vector $\vec{n}\begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$ is normal to the plane (CFI). c. Verify that a Cartesian equation of the plane (CFI) is: $x + 2y + 2z - 3 = 0$.
2. Let $d$ be the line passing through G and perpendicular to the plane (CFI). a. Determine a parametric representation of the line $d$. b. Prove that the point $\mathrm{K}\left(\frac{7}{9}; \frac{5}{9}; \frac{5}{9}\right)$ is the orthogonal projection of point G onto the plane (CFI). c. Deduce from the previous questions that the distance from point G to the plane (CFI) is equal to $\frac{2}{3}$.
3. Consider the pyramid GCFI. Recall that the volume $V$ of a pyramid is given by the formula $$V = \frac{1}{3} \times b \times h$$ where $b$ is the area of a base and $h$ is the height associated with this base. a. Prove that the volume of the pyramid GCFI is equal to $\frac{1}{6}$, expressed in cubic units. b. Deduce the area of triangle CFI, in square units.

Space is referred to an orthonormal coordinate system in which we consider:

• the points $\mathrm{A}(2;-1;0)$, $\mathrm{B}(1;0;-3)$, $\mathrm{C}(6;6;1)$ and $\mathrm{E}(1;2;4)$;
• The plane $\mathscr{P}$ with Cartesian equation $2x - y - z + 4 = 0$.

1. a. Prove that triangle ABC is right-angled at A. b. Calculate the dot product $\overrightarrow{\mathrm{BA}} \cdot \overrightarrow{\mathrm{BC}}$ then the lengths BA and BC. c. Deduce the measure in degrees of the angle $\widehat{\mathrm{ABC}}$ rounded to the nearest degree.
2. a. Prove that the plane $\mathscr{P}$ is parallel to the plane ABC. b. Deduce a Cartesian equation of the plane ABC. c. Determine a parametric representation of the line $\mathscr{D}$ orthogonal to the plane ABC and passing through point E. d. Prove that the orthogonal projection H of point E onto the plane ABC has coordinates $\left(4; \frac{1}{2}; \frac{5}{2}\right)$.
3. Recall that the volume of a pyramid is given by $V = \frac{1}{3}\mathscr{B}h$ where $\mathscr{B}$ denotes the area of a base and $h$ the height of the pyramid associated with this base. Calculate the area of triangle ABC then prove that the volume of the pyramid ABCE is equal to $16.5$ cubic units.

Consider the cube ABCDEFGH with edge length 1. The space is equipped with the orthonormal frame $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$. Point I is the midpoint of segment $[\mathrm{EF}]$, K is the center of square ADHE, and O is the midpoint of segment [AG].
The goal of the exercise is to calculate in two different ways the distance from point B to the plane (AIG).

Part 1. First method

1. Give, without justification, the coordinates of points $\mathrm{A}$, $\mathrm{B}$, and G.

We admit that points I and K have coordinates $\mathrm{I}\left(\frac{1}{2}; 0; 1\right)$ and $\mathrm{K}\left(0; \frac{1}{2}; \frac{1}{2}\right)$.

1. Prove that the line (BK) is orthogonal to the plane (AIG).
2. Verify that a Cartesian equation of the plane (AIG) is: $2x - y - z = 0$.
3. Give a parametric representation of the line (BK).
4. Deduce that the orthogonal projection L of point B onto the plane (AIG) has coordinates $\mathrm{L}\left(\frac{1}{3}; \frac{1}{3}; \frac{1}{3}\right)$.
5. Determine the distance from point B to the plane (AIG).

Part 2. Second method

Recall that the volume $V$ of a pyramid is given by the formula $V = \frac{1}{3} \times b \times h$, where $b$ is the area of a base and $h$ is the height associated with this base.

1. a. Justify that in the tetrahedron $\mathrm{ABIG}$, $[\mathrm{GF}]$ is the height relative to the base AIB. b. Deduce the volume of the tetrahedron ABIG.
2. We admit that $\mathrm{AI} = \mathrm{IG} = \frac{\sqrt{5}}{2}$ and that $\mathrm{AG} = \sqrt{3}$. Prove that the area of the isosceles triangle AIG is equal to $\frac{\sqrt{6}}{4}$ square units.
3. Deduce the distance from point B to the plane (AIG).

Consider the right prism ABFEDCGH, with base ABFE, a right trapezoid at A. We associate with this prism the orthonormal coordinate system $(A; \vec{\imath}, \vec{\jmath}, \vec{k})$ such that:
$$\vec{\imath} = \frac{1}{4}\overrightarrow{AB}, \quad \vec{\jmath} = \frac{1}{4}\overrightarrow{AD}, \quad \vec{k} = \frac{1}{8}\overrightarrow{AE}$$
Moreover we have $\overrightarrow{BF} = \frac{1}{2}\overrightarrow{AE}$. We denote I the midpoint of segment $[EF]$. We denote J the midpoint of segment $[AE]$.

1. Give the coordinates of points I and J.
2. Let $\vec{n}$ be the vector with coordinates $\left(\begin{array}{c} -1 \\ 1 \\ 1 \end{array}\right)$. a. Show that the vector $\vec{n}$ is normal to the plane (IGJ). b. Determine a Cartesian equation of the plane (IGJ).
3. Determine a parametric representation of the line $d$, perpendicular to the plane (IGJ) and passing through H.
4. We denote L the orthogonal projection of point H onto the plane (IGJ). Show that the coordinates of L are $\left(\frac{8}{3}; \frac{4}{3}; \frac{16}{3}\right)$.
5. Calculate the distance from point H to the plane (IGJ).
6. Show that triangle IGJ is right-angled at I.
7. Deduce the volume of tetrahedron IGJH.

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

The figure below corresponds to the model of an architectural project. It is a house with a cubic shape (ABCDEFGH) attached to a garage with a cubic shape (BIJKLMNO) where L is the midpoint of segment [BF] and K is the midpoint of segment [BC]. The garage is topped with a roof with a pyramidal shape (LMNOP) with square base LMNO and apex P positioned on the facade of the house.
We equip space with the orthonormal coordinate system $(A; \vec{\imath}, \vec{\jmath}, \vec{k})$, with $\vec{\imath} = \frac{1}{2}\overrightarrow{AB}$, $\vec{\jmath} = \frac{1}{2}\overrightarrow{AD}$ and $\vec{k} = \frac{1}{2}\overrightarrow{AE}$.

1. a. By reading the graph, give the coordinates of points H, M and N. b. Determine a parametric representation of the line (HM).
2. The architect places point P at the intersection of line (HM) and plane (BCF). Show that the coordinates of P are $\left(2; \frac{2}{3}; \frac{4}{3}\right)$.
3. a. Calculate the dot product $\overrightarrow{PM} \cdot \overrightarrow{PN}$. b. Calculate the distance PM. We admit that the distance PN is equal to $\frac{\sqrt{11}}{3}$. c. To satisfy technical constraints, the roof can only be built if the angle $\widehat{MPN}$ does not exceed $55°$. Can the roof be built?
4. Justify that the lines (HM) and (EN) are secant. What is their point of intersection?

Space is referred to an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. Consider the points $$\mathrm{A}(1;0;-1), \quad \mathrm{B}(3;-1;2), \quad \mathrm{C}(2;-2;-1) \quad \text{and} \quad \mathrm{D}(4;-1;-2).$$ We denote by $\Delta$ the line with parametric representation $$\left\{\begin{aligned} x &= 0 \\ y &= 2+t, \text{ with } t \in \mathbb{R}. \\ z &= -1+t \end{aligned}\right.$$

1. a. Show that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ define a plane which we will denote $\mathscr{P}$. b. Show that the line (CD) is orthogonal to the plane $\mathscr{P}$. On the plane $\mathscr{P}$, what does point C represent with respect to D? c. Show that a Cartesian equation of the plane $\mathscr{P}$ is: $2x + y - z - 3 = 0$.
2. a. Calculate the distance CD. b. Does there exist a point M on the plane $\mathscr{P}$ different from C satisfying $\mathrm{MD} = \sqrt{6}$? Justify your answer.
3. a. Show that the line $\Delta$ is contained in the plane $\mathscr{P}$. Let H be the orthogonal projection of point D onto the line $\Delta$. b. Show that H is the point of $\Delta$ associated with the value $t = -2$ in the parametric representation of $\Delta$ given above. c. Deduce the distance from point D to the line $\Delta$.

We consider the cube ABCDEFGH with edge 1. We call I the point of intersection of the plane (GBD) with the line (EC). The space is referred to the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$.

1. Give in this coordinate system the coordinates of points $\mathrm{E}, \mathrm{C}, \mathrm{G}$.
2. Determine a parametric representation of the line (EC).
3. Prove that the line (EC) is orthogonal to the plane (GBD).
4. 1. [a.] Justify that a Cartesian equation of the plane (GBD) is: $$x + y - z - 1 = 0.$$
   2. [b.] Show that point I has coordinates $\left(\frac{2}{3}; \frac{2}{3}; \frac{1}{3}\right)$.
   3. [c.] Deduce that the distance from point E to the plane (GBD) is equal to $\frac{2\sqrt{3}}{3}$.
5. 1. [a.] Prove that triangle BDG is equilateral.
   2. [b.] Calculate the area of triangle BDG. You may use point J, the midpoint of segment [BD].
6. Justify that the volume of tetrahedron EGBD is equal to $\frac{1}{3}$.
   We recall that the volume of a tetrahedron is given by $V = \frac{1}{3}Bh$ where $B$ is the area of a base of the tetrahedron and $h$ is the height relative to this base.

We consider two cubes ABCDEFGH and BKLCFJMG positioned as in the following figure. The point I is the midpoint of [EF]. We place ourselves in the orthonormal coordinate system $(A; \overrightarrow{AB}; \overrightarrow{AD}; \overrightarrow{AE})$. The points F, G and J have coordinates $$\mathrm{F}(1;0;1), \quad \mathrm{G}(1;1;1) \quad \text{and} \quad \mathrm{J}(2;0;1).$$

1. Show that the volume of the tetrahedron FIGB is equal to $\frac{1}{12}$ unit of volume.
   Recall that the volume $V$ of a tetrahedron is given by the formula: $$V = \frac{1}{3} \times \text{area of a base} \times \text{corresponding height.}$$
   
2. Determine the coordinates of point I.
3. Show that the vector $\overrightarrow{\mathrm{DJ}}$ is a normal vector to the plane (BIG).
4. Show that a Cartesian equation of the plane (BIG) is $2x - y + z - 2 = 0$.
5. Determine a parametric representation of the line $d$, perpendicular to (BIG) and passing through F.
6. a. The line $d$ intersects the plane (BIG) at point $\mathrm{L}'$. Show that the coordinates of point $\mathrm{L}'$ are $\left(\frac{2}{3}; \frac{1}{6}; \frac{5}{6}\right)$. b. Calculate the length $\mathrm{FL}'$. c. Deduce from the previous questions the area of triangle IGB.

We consider the cube ABCDEFGH with edge length 1 represented opposite. We denote K the midpoint of segment [HG]. We place ourselves in the orthonormal coordinate system $( \mathrm { A } ; \overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AE } } )$.

1. Justify that the points $\mathrm { C } , \mathrm { F }$ and K define a plane.
2. a. Give, without justification, the lengths KG, GF and GC. b. Calculate the area of triangle FGC. c. Calculate the volume of tetrahedron FGCK.

We recall that the volume $V$ of a tetrahedron 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.
3. a. We denote $\vec { n }$ the vector with coordinates $\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right)$.
Prove that $\vec { n }$ is normal to the plane (CFK). b. Deduce that a Cartesian equation of the plane (CFK) is: $$x + 2 y + z - 3 = 0 .$$

1. We denote $\Delta$ the line passing through point G and perpendicular to the plane (CFK). Prove that a parametric representation of the line $\Delta$ is:

$$\left\{ \begin{aligned} x & = 1 + t \\ y & = 1 + 2 t \\ z & = 1 + t \end{aligned} \quad ( t \in \mathbb { R } ) \right)$$

1. Let L be the point of intersection between the line $\Delta$ and the plane (CFK). a. Determine the coordinates of point L . b. Deduce that $\mathrm { LG } = \frac { \sqrt { 6 } } { 6 }$.
2. Using question 2., determine the exact value of the area of triangle CFK.

In an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$ we consider the points: $$\mathrm{A}(1;1;-4), \quad \mathrm{B}(2;-1;-3), \quad \mathrm{C}(0;-1;-1) \text{ and } \Omega(1;1;2).$$

1. Prove that the points $\mathrm{A}$, $\mathrm{B}$ and C define a plane.
2. a. Prove that the vector $\vec{n}$ with coordinates $\left(\begin{array}{l}1\\1\\1\end{array}\right)$ is normal to the plane (ABC). b. Justify that a Cartesian equation of the plane (ABC) is $x + y + z + 2 = 0$.
3. a. Justify that the point $\Omega$ does not belong to the plane (ABC). b. Determine the coordinates of the point H, the orthogonal projection of the point $\Omega$ onto the plane (ABC).

We admit that $\Omega\mathrm{H} = 2\sqrt{3}$. We define the sphere $S$ with centre $\Omega$ and radius $2\sqrt{3}$ as the set of all points M in space such that $\Omega\mathrm{M} = 2\sqrt{3}$.
4. Justify, without calculation, that any point N of the plane (ABC), distinct from H, does not belong to the sphere $S$. We say that a plane $\mathscr{P}$ is tangent to the sphere $S$ at a point K when the following two conditions are satisfied:

• $\mathrm{K} \in \mathscr{P} \cap S$
• $(\Omega\mathrm{K}) \perp \mathscr{P}$

1. Let the plane $\mathscr{P}$ with Cartesian equation $x + y - z - 6 = 0$ and the point K with coordinates $\mathrm{K}(3;3;0)$. Prove that the plane $\mathscr{P}$ is tangent to the sphere $S$ at point K.
2. We admit that the planes (ABC) and $\mathscr{P}$ intersect along a line ($\Delta$). Determine a parametric equation of the line ($\Delta$).

We consider the cube ABCDEFGH which is represented in APPENDIX. In the orthonormal coordinate system ( $A$; $\overrightarrow { A B }$; $\overrightarrow { A D }$; $\overrightarrow { A E }$ ), we consider the points $M , N$ and $P$ with coordinates:
$$\mathrm { M } \left( 1 ; 1 ; \frac { 3 } { 4 } \right) , \quad \mathrm { N } \left( 0 ; \frac { 1 } { 2 } ; 1 \right) , \quad \mathrm { P } \left( 1 ; 0 ; - \frac { 5 } { 4 } \right)$$
In this exercise, we propose to calculate the volume of the tetrahedron FMNP.

1. Give the coordinates of the vectors $\overrightarrow { \mathrm { MN } }$ and $\overrightarrow { \mathrm { MP } }$.
2. Place the points $\mathrm { M} , \mathrm { N }$ and P on the figure given in APPENDIX which must be returned with your work.
3. Justify that the points $\mathrm { M } , \mathrm { N }$ and P are not collinear.

From then on, the three points define the plane (MNP).
4. a. Calculate the dot product $\overrightarrow { \mathrm { MN } } \cdot \overrightarrow { \mathrm { MP } }$, then deduce the nature of the triangle MNP. b. Calculate the area of the triangle MNP.
5. a. Show that the vector $\vec { n } ( 5 ; - 8 ; 4 )$ is a normal vector to the plane (MNP). b. Deduce that a Cartesian equation of the plane (MNP) is $5 x - 8 y + 4 z = 0$. 6. We recall that the point F has coordinates $\mathrm { F } ( 1 ; 0 ; 1 )$.
Determine a parametric representation of the line $d$ orthogonal to the plane (MNP) and passing through the point F. 7. We denote L the orthogonal projection of the point F onto the plane (MNP).
Show that the coordinates of the point L are: $\mathrm { L } \left( \frac { 4 } { 7 } ; \frac { 24 } { 35 } ; \frac { 23 } { 35 } \right)$. 8. Show that $\mathrm { FL } = \frac { 3 \sqrt { 105 } } { 35 }$ then calculate the volume of the tetrahedron FMNP.
We recall that the volume V of a tetrahedron is given by the formula:
$$V = \frac { 1 } { 3 } \times \text { area of a base } \times \text{ height associated with this base. }$$

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

• $d_1$ the line passing through point $H(2; 3; 0)$ with direction vector $\vec{u}\left(\begin{array}{c}1\\-1\\1\end{array}\right)$;
• $d_2$ the line with parametric representation:

$$\left\{\begin{aligned} x &= 2k - 3\\ y &= k\\ z &= 5 \end{aligned}\quad\text{where }k\text{ describes }\mathbb{R}.\right.$$ The purpose of this exercise is to determine a parametric representation of a line $\Delta$ that is perpendicular to both lines $d_1$ and $d_2$.

1. a. Determine a direction vector $\vec{v}$ of line $d_2$. b. Prove that lines $d_1$ and $d_2$ are not parallel. c. Prove that lines $d_1$ and $d_2$ are not intersecting. d. What is the relative position of lines $d_1$ and $d_2$?
2. a. Verify that the vector $\vec{w}\left(\begin{array}{c}-1\\2\\3\end{array}\right)$ is orthogonal to both $\vec{u}$ and $\vec{v}$. b. We consider the plane $P$ passing through point $H$ and directed by vectors $\vec{u}$ and $\vec{w}$. We admit that a Cartesian equation of this plane is: $$5x + 4y - z - 22 = 0.$$ Prove that the intersection of plane $P$ and line $d_2$ is the point $M(3; 3; 5)$.
3. Let $\Delta$ be the line with direction vector $\vec{w}$ passing through point $M$.
   A parametric representation of $\Delta$ is therefore given by: $$\left\{\begin{array}{l} x = -r + 3\\ y = 2r + 3\\ z = 3r + 5 \end{array}\text{ where }r\text{ describes }\mathbb{R}.\right.$$ a. Justify that lines $\Delta$ and $d_1$ are perpendicular at a point $L$ whose coordinates you will determine. b. Explain why line $\Delta$ is a solution to the problem posed.

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

• the point $A(1; -1; -1)$;
• the plane $\mathscr{P}_{1}$, with equation: $5x + 2y + 4z = 17$;
• the plane $\mathscr{P}_{2}$ with equation: $10x + 14y + 3z = 19$;
• the line $\mathscr{D}$ with parametric representation: $$\left\{ \begin{aligned} x & = 1 + 2t \\ y & = -t \\ z & = 3 - 2t \end{aligned} \text{ where } t \text{ ranges over } \mathbb{R} . \right.$$

1. Justify that the planes $\mathscr{P}_{1}$ and $\mathscr{P}_{2}$ are not parallel.
2. Prove that $\mathscr{D}$ is the line of intersection of $\mathscr{P}_{1}$ and $\mathscr{P}_{2}$.
3. a. Verify that A does not belong to $\mathscr{P}_{1}$. b. Justify that A does not belong to $\mathscr{D}$.
4. For every real $t$, we denote $M$ the point of $\mathscr{D}$ with coordinates $(1 + 2t; -t; 3 - 2t)$. We then consider the function $f$ which associates to every real $t$ the value $AM^{2}$, that is $f(t) = AM^{2}$. a. Prove that for every real $t$, we have: $f(t) = 9t^{2} - 18t + 17$. b. Prove that the distance AM is minimal when $M$ has coordinates $(3; -1; 1)$.
5. We denote H the point with coordinates $(3; -1; 1)$. Prove that the line (AH) is perpendicular to $\mathscr{D}$.

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 to a question neither awards nor deducts points. The five questions are independent. Space is equipped with an orthonormal reference frame $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider the points $\mathrm{A}(-1; 2; 5)$, $\mathrm{B}(3; 6; 3)$, $\mathrm{C}(3; 0; 9)$ and $\mathrm{D}(8; -3; -8)$. We admit that points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are not collinear.

1. Triangle ABC is: a. isosceles right-angled at A b. isosceles right-angled at B c. isosceles right-angled at C d. equilateral
   
2. A Cartesian equation of plane (BCD) is: a. $2x + y + z - 15 = 0$ b. $9x - 5y + 3 = 0$ c. $4x + y + z - 21 = 0$ d. $11x + 5z - 73 = 0$
   
3. We admit that plane $(\mathrm{ABC})$ has Cartesian equation $x - 2y - 2z + 15 = 0$. We call H the orthogonal projection of point D onto plane (ABC). We can affirm that: a. $\mathrm{H}(-2; 17; 12)$ b. $\mathrm{H}(3; 7; 2)$ c. $\mathrm{H}(3; 2; 7)$ d. $\mathrm{H}(-15; 1; -1)$
   
4. Let the line $\Delta$ with parametric representation $\left\{\begin{array}{l} x = 5 + t \\ y = 3 - t \\ z = -1 + 3t \end{array}\right.$, with $t$ real. Lines (BC) and $\Delta$ are: a. coincident b. strictly parallel c. intersecting d. non-coplanar
   
5. We consider the plane $\mathscr{P}$ with Cartesian equation $2x - y + 2z - 6 = 0$. We admit that plane (ABC) has Cartesian equation $x - 2y - 2z + 15 = 0$. We can affirm that: a. planes $\mathscr{P}$ and (ABC) are strictly parallel b. planes $\mathscr{P}$ and (ABC) are intersecting and their intersection is line (AB) c. planes $\mathscr{P}$ and (ABC) are intersecting and their intersection is line (AC) d. planes $\mathscr{P}$ and (ABC) are intersecting and their intersection is line (BC)

In space equipped with an orthonormal coordinate system with unit 1 cm, we consider the points
$$\mathrm{D}(3;1;5), \quad \mathrm{E}(3;-2;-1), \quad \mathrm{F}(-1;2;1), \quad \mathrm{G}(3;2;-3).$$

1. a. Determine the coordinates of the vectors $\overrightarrow{\mathrm{EF}}$ and $\overrightarrow{\mathrm{FG}}$. b. Justify that the points $\mathrm{E}$, $\mathrm{F}$ and $\mathrm{G}$ are not collinear.
2. a. Determine a parametric representation of the line (FG). b. We call H the point with coordinates $(2; 2; -2)$. Verify that H is the orthogonal projection of E onto the line (FG). c. Show that the area of triangle EFG is equal to $12\text{ cm}^{2}$.
3. a. Prove that the vector $\vec{n}\begin{pmatrix}2\\1\\2\end{pmatrix}$ is a normal vector to the plane (EFG). b. Determine a Cartesian equation of the plane (EFG). c. Determine a parametric representation of the line $(d)$ passing through point D and orthogonal to the plane (EFG). d. We denote K the orthogonal projection of point D onto the plane (EFG). Using the previous questions, calculate the coordinates of point K.
4. a. Verify that the distance $DK$ is equal to 5 cm. b. Deduce the volume of the tetrahedron DEFG.

In space with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ), we consider the points
$$\mathrm { A } ( 0 ; 4 ; 16 ) , \quad \mathrm { B } ( 0 ; 4 ; - 10 ) , \quad \mathrm { C } ( 4 ; - 8 ; 0 ) \quad \text { and } \quad \mathrm { K } ( 0 ; 4 ; 3 ) .$$
We define the sphere $S$ with center K and radius 13 as the set of points M such that $\mathrm { KM } = 13$.

1. a. Verify that point C belongs to sphere $S$. b. Show that triangle ABC is right-angled at C.
2. a. Show that the vector $\vec { n } \left( \begin{array} { l } 3 \\ 1 \\ 0 \end{array} \right)$ is a normal vector to plane (ABC). b. Determine a Cartesian equation of plane (ABC).
3. We admit that sphere $S$ intersects the x-axis at two points, one having a positive abscissa and the other a negative abscissa. We denote D the one with positive abscissa. a. Show that point D has coordinates $( 12 ; 0 ; 0 )$. b. Give a parametric representation of the line $\Delta$ passing through D and perpendicular to plane (ABC). c. Determine the distance from point D to plane (ABC).
4. Calculate an approximate value, to the nearest unit of volume, of the volume of tetrahedron ABCD. We recall the formula for the volume V of a tetrahedron $$V = \frac { 1 } { 3 } \times \mathscr { B } \times h$$ where $\mathscr { B }$ is the area of a base and h the associated height.

Space is equipped with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ). Consider the points
$$\mathrm { A } ( 3 ; 0 ; 1 ) , \quad \mathrm { B } ( 2 ; 1 ; 2 ) \text { and } \quad \mathrm { C } ( - 2 ; - 5 ; 1 ) .$$

1. Prove that the points $\mathrm { A } , \mathrm { B }$ and C are not collinear.
2. Prove that the triangle ABC is right-angled at A .
3. Verify that the plane $( \mathrm { ABC } )$ has the Cartesian equation :

$$- x + y - 2 z + 5 = 0$$

1. Consider the point $S ( 1 ; - 2 ; 4 )$.

Determine the parametric representation of the line ( $\Delta$ ), passing through S and orthogonal to the plane (ABC).
5. We call H the point of intersection of the line ( $\Delta$ ) and the plane (ABC).
Show that the coordinates of H are $( 0 ; - 1 ; 2 )$. 6. Calculate the exact value of the distance SH. 7. Consider the circle $\mathscr { C }$, included in the plane (ABC), with center H, passing through the point B. We call $\mathscr { D }$ the disk bounded by the circle $\mathscr { C }$.
Determine the exact value of the area of the disk $\mathscr { D }$. 8. Deduce the exact value of the volume of the cone with apex S and base the disk $\mathscr { D }$.

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.

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