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 $\mathscr{D}$.
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 $\overrightarrow{AB} \cdot \vec{u}$.
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.
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.
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 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:
\begin{itemize}
\item the point A with coordinates $(-1; 1; 3)$,
\item the line $\mathscr{D}$ whose parametric representation is: $\left\{\begin{aligned} x &= 1 + 2t \\ y &= 2 - t \\ z &= 2 + 2t \end{aligned} \quad t \in \mathbb{R}\right.$.
\end{itemize}
It is admitted that point A does not belong to line $\mathscr{D}$.
\begin{enumerate}
\item 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 $\overrightarrow{AB} \cdot \vec{u}$.
\item 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.
\item 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.
\item 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.
\end{enumerate}