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

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

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

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

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

Exercise 3 — 7 points Theme: Geometry in space Space is equipped with an orthonormal coordinate system $(\mathrm{O};\vec{\imath},\vec{\jmath},\vec{k})$. We consider the points $$\mathrm{A}(3;-2;2), \quad \mathrm{B}(6;1;5), \quad \mathrm{C}(6;-2;-1) \quad \text{and} \quad \mathrm{D}(0;4;-1).$$ We recall that the volume of a tetrahedron is given by the formula: $$V = \frac{1}{3}\mathscr{A} \times h$$ where $\mathscr{A}$ is the area of the base and $h$ is the corresponding height.

1. Prove that the points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$ and $\mathrm{D}$ are not coplanar.
   
2. 1. [a.] Show that the triangle ABC is right-angled.
   2. [b.] Show that the line (AD) is perpendicular to the plane (ABC).
   3. [c.] Deduce the volume of the tetrahedron ABCD.
   
   
3. We consider the point $\mathrm{H}(5;0;1)$.
   1. [a.] Show that there exist real numbers $\alpha$ and $\beta$ such that $\overrightarrow{\mathrm{BH}} = \alpha\overrightarrow{\mathrm{BC}} + \beta\overrightarrow{\mathrm{BD}}$.
   2. [b.] Prove that H is the orthogonal projection of point A onto the plane (BCD).
   3. [c.] Deduce the distance from point A to the plane (BCD).
   
   
4. Deduce from the previous questions the area of triangle BCD.

Exercise 4 — 6 points

Theme: Exponential function Main topics covered: Geometry in space The space is equipped with an orthonormal coordinate system $( \mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k } )$. We consider the points $\mathrm { A } ( 5 ; 0 ; - 1 ) , \mathrm { B } ( 1 ; 4 ; - 1 ) , \mathrm { C } ( 1 ; 0 ; 3 ) , \mathrm { D } ( 5 ; 4 ; 3 )$ and $\mathrm { E } ( 10 ; 9 ; 8 )$.

1. a. Let R be the midpoint of the segment $[ \mathrm { AB } ]$. Calculate the coordinates of point R as well as the coordinates of the vector $\overrightarrow { \mathrm { AB } }$. b. Let $\mathscr { P } _ { 1 }$ be the plane passing through point R and for which $\overrightarrow { \mathrm { AB } }$ is a normal vector. Prove that a Cartesian equation of the plane $\mathscr { P } _ { 1 }$ is: $$x - y - 1 = 0 .$$ c. Prove that point E belongs to the plane $\mathscr { P } _ { 1 }$ and that $\mathrm { EA } = \mathrm { EB }$.
   
2. We consider the plane $\mathscr { P } _ { 2 }$ with Cartesian equation $x - z - 2 = 0$. a. Justify that the planes $\mathscr { P } _ { 1 }$ and $\mathscr { P } _ { 2 }$ are secant. b. We denote $\Delta$ the line of intersection of $\mathscr { P } _ { 1 }$ and $\mathscr { P } _ { 2 }$. Prove that a parametric representation of the line $\Delta$ is: $$\left\{ \begin{aligned} x & = 2 + t \\ y & = 1 + t \quad ( t \in \mathbb { R } ) . \\ z & = t \end{aligned} \right.$$
   
3. We consider the plane $\mathscr { P } _ { 3 }$ with Cartesian equation $y + z - 3 = 0$. Justify that the line $\Delta$ is secant to the plane $\mathscr { P } _ { 3 }$ at a point $\Omega$ whose coordinates you will determine.
   If S and T are two distinct points in space, we recall that the set of points M in space such that $\mathrm{MS} = \mathrm{MT}$ is a plane, called the perpendicular bisector plane of the segment $[ \mathrm { ST } ]$. We assume that the planes $\mathscr { P } _ { 1 }$, $\mathscr { P } _ { 2 }$ and $\mathscr { P } _ { 3 }$ are the perpendicular bisector planes of the segments [AB], [AC] and [AD] respectively.
   
4. a. Justify that $\Omega A = \Omega B = \Omega C = \Omega D$. b. Deduce that the points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D belong to the same sphere, whose centre and radius you will specify.

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

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

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

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

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

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

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

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

Exercise 3 (7 points) Theme: geometry in space Consider a cube ABCDEFGH and call K the midpoint of segment [BC]. We place ourselves in the coordinate system $(A; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$ and consider the tetrahedron EFGK. Recall that the volume of a tetrahedron is given by: $$V = \frac{1}{3} \times \mathscr{B} \times h$$ where $\mathscr{B}$ denotes the area of a base and $h$ the height relative to this base.

1. Specify the coordinates of points $\mathrm{E}, \mathrm{F}, \mathrm{G}$ and K.
2. Show that the vector $\vec{n}\left(\begin{array}{r}2\\-2\\1\end{array}\right)$ is orthogonal to the plane (EGK).
3. Prove that the plane (EGK) has the Cartesian equation: $2x - 2y + z - 1 = 0$.
4. Determine a parametric representation of the line (d) orthogonal to the plane (EGK) passing through F.
5. Show that the orthogonal projection $L$ of $F$ onto the plane (EGK) has coordinates $\left(\frac{5}{9}; \frac{4}{9}; \frac{7}{9}\right)$.
6. Justify that the length LF is equal to $\frac{2}{3}$.
7. Calculate the area of triangle EFG. Deduce that the volume of tetrahedron EFGK is equal to $\frac{1}{6}$.
8. Deduce from the previous questions the area of triangle EGK.
9. Consider the points P midpoint of segment [EG], M midpoint of segment [EK] and N midpoint of segment [GK]. Determine the volume of tetrahedron FPMN.

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).

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.

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.

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?

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 $\mathscr { P }$.

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 $\mathscr { D }$.

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.

In space with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider:

• the plane $\mathscr{P}_1$ whose Cartesian equation is $2x + y - z + 2 = 0$,
• the plane $\mathscr{P}_2$ passing through point $\mathrm{B}(1; 1; 2)$ and whose normal vector is $\overrightarrow{n_2}\left(\begin{array}{c}1\\-1\\1\end{array}\right)$.

1. a. Give the coordinates of a vector $\overrightarrow{n_1}$ normal to the plane $\mathscr{P}_1$. b. We recall that two planes are perpendicular if a normal vector to one of the planes is orthogonal to a normal vector to the other plane. Show that the planes $\mathscr{P}_1$ and $\mathscr{P}_2$ are perpendicular.
2. a. Determine a Cartesian equation of the plane $\mathscr{P}_2$. b. We denote by $\Delta$ the line whose parametric representation is: $$\left\{\begin{array}{rl} x &= 0 \\ y &= -2 + t \\ z &= t \end{array},\quad t \in \mathbb{R}\right.$$ Show that the line $\Delta$ is the intersection of the planes $\mathscr{P}_1$ and $\mathscr{P}_2$.
3. We consider the point $\mathrm{A}(1; 1; 1)$ and we admit that point A belongs to neither $\mathscr{P}_1$ nor $\mathscr{P}_2$. We denote by H the orthogonal projection of point A onto the line $\Delta$. We recall that, from question 2.b, the line $\Delta$ is the set of points $M_t$ with coordinates $(0; -2+t; t)$, where $t$ denotes any real number. a. Show that, for every real $t$, $\mathrm{A}M_t = \sqrt{2t^2 - 8t + 11}$. b. Deduce that $\mathrm{AH} = \sqrt{3}$.
4. We denote by $\mathscr{D}_1$ the line perpendicular to the plane $\mathscr{P}_1$ passing through point A and $\mathrm{H}_1$ the orthogonal projection of point A onto the plane $\mathscr{P}_1$. a. Determine a parametric representation of the line $\mathscr{D}_1$. b. Deduce that the point $\mathrm{H}_1$ has coordinates $\left(-\frac{1}{3}; \frac{1}{3}; \frac{5}{3}\right)$.
5. Let $\mathrm{H}_2$ be the orthogonal projection of A onto the plane $\mathscr{P}_2$. We admit that $\mathrm{H}_2$ has coordinates $\left(\frac{4}{3}; \frac{2}{3}; \frac{4}{3}\right)$ and that H has coordinates $(0; 0; 2)$. Show that $\mathrm{AH}_1\mathrm{HH}_2$ is a rectangle.

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?