Main topics covered: Space geometry with respect to an orthonormal coordinate system; orthogonality in space In an orthonormal coordinate system $( \mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k } )$ we consider
the point A with coordinates $( 1 ; 3 ; 2 )$,
the vector $\vec { u }$ with coordinates $\left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right)$
the line $d$ passing through the origin O of the coordinate system and having $\vec { u }$ as its direction vector.
The purpose of this exercise is to determine the point on $d$ closest to point A and to study some properties of this point.
Determine a parametric representation of the line $d$.
Let $t$ be any real number, and $M$ a point on the line $d$, the point $M$ having coordinates $( t ; t ; 0 )$. a. We denote $AM$ the distance between points A and $M$. Prove that: $$AM ^ { 2 } = 2 t ^ { 2 } - 8 t + 14 .$$ b. Prove that the point $M _ { 0 }$ with coordinates $( 2 ; 2 ; 0 )$ is the point on the line $d$ for which the distance $AM$ is minimal. We will assume that the distance $AM$ is minimal when its square $AM ^ { 2 }$ is minimal.
Prove that the lines $( A M _ { 0 } )$ and $d$ are orthogonal.
We call $A ^ { \prime }$ the orthogonal projection of point $A$ onto the plane with Cartesian equation $z = 0$. The point $A ^ { \prime }$ therefore has coordinates $( 1 ; 3 ; 0 )$. Prove that the point $M _ { 0 }$ is the point of the plane $\left( A A ^ { \prime } M _ { 0 } \right)$ closest to point O, the origin of the coordinate system.
Calculate the volume of the pyramid $O M _ { 0 } A ^ { \prime } A$. Recall that the volume of a pyramid is given by: $V = \frac { 1 } { 3 } \mathscr { B } h$, where $\mathscr { B }$ is the area of a base and $h$ is the height of the pyramid corresponding to this base.
Main topics covered: Space geometry with respect to an orthonormal coordinate system; orthogonality in space
In an orthonormal coordinate system $( \mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k } )$ we consider
\begin{itemize}
\item the point A with coordinates $( 1 ; 3 ; 2 )$,
\item the vector $\vec { u }$ with coordinates $\left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right)$
\item the line $d$ passing through the origin O of the coordinate system and having $\vec { u }$ as its direction vector.
\end{itemize}
The purpose of this exercise is to determine the point on $d$ closest to point A and to study some properties of this point.
\begin{enumerate}
\item Determine a parametric representation of the line $d$.
\item Let $t$ be any real number, and $M$ a point on the line $d$, the point $M$ having coordinates $( t ; t ; 0 )$.\\
a. We denote $AM$ the distance between points A and $M$. Prove that:
$$AM ^ { 2 } = 2 t ^ { 2 } - 8 t + 14 .$$
b. Prove that the point $M _ { 0 }$ with coordinates $( 2 ; 2 ; 0 )$ is the point on the line $d$ for which the distance $AM$ is minimal.\\
We will assume that the distance $AM$ is minimal when its square $AM ^ { 2 }$ is minimal.
\item Prove that the lines $( A M _ { 0 } )$ and $d$ are orthogonal.
\item We call $A ^ { \prime }$ the orthogonal projection of point $A$ onto the plane with Cartesian equation $z = 0$. The point $A ^ { \prime }$ therefore has coordinates $( 1 ; 3 ; 0 )$.
Prove that the point $M _ { 0 }$ is the point of the plane $\left( A A ^ { \prime } M _ { 0 } \right)$ closest to point O, the origin of the coordinate system.
\item Calculate the volume of the pyramid $O M _ { 0 } A ^ { \prime } A$.
Recall that the volume of a pyramid is given by: $V = \frac { 1 } { 3 } \mathscr { B } h$, where $\mathscr { B }$ is the area of a base and $h$ is the height of the pyramid corresponding to this base.
\end{enumerate}