In space equipped with the orthonormal coordinate system ($\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k}$) with unit 1 cm, we consider the points $\mathrm{A}$, $\mathrm{B}$, C and D with coordinates respectively $(2; 1; 4)$, $(4; -1; 0)$, $(0; 3; 2)$ and $(4; 3; -2)$.
Determine a parametric representation of the line (CD).
Let $M$ be a point on the line (CD). a. Determine the coordinates of the point $M$ such that the distance $BM$ is minimal. b. We denote H the point on the line $(\mathrm{CD})$ with coordinates $(3; 3; -1)$. Verify that the lines $(\mathrm{BH})$ and $(\mathrm{CD})$ are perpendicular. c. Show that the area of triangle BCD is equal to $12\,\mathrm{cm}^2$.
a. Prove that the vector $\vec{n}\begin{pmatrix}2\\1\\2\end{pmatrix}$ is a normal vector to the plane (BCD). b. Determine a Cartesian equation of the plane (BCD).
In space equipped with the orthonormal coordinate system ($\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k}$) with unit 1 cm, we consider the points $\mathrm{A}$, $\mathrm{B}$, C and D with coordinates respectively $(2; 1; 4)$, $(4; -1; 0)$, $(0; 3; 2)$ and $(4; 3; -2)$.
\begin{enumerate}
\item Determine a parametric representation of the line (CD).
\item Let $M$ be a point on the line (CD).\\
a. Determine the coordinates of the point $M$ such that the distance $BM$ is minimal.\\
b. We denote H the point on the line $(\mathrm{CD})$ with coordinates $(3; 3; -1)$. Verify that the lines $(\mathrm{BH})$ and $(\mathrm{CD})$ are perpendicular.\\
c. Show that the area of triangle BCD is equal to $12\,\mathrm{cm}^2$.
\item a. Prove that the vector $\vec{n}\begin{pmatrix}2\\1\\2\end{pmatrix}$ is a normal vector to the plane (BCD).\\
b. Determine a Cartesian equation of the plane (BCD).
\end{enumerate}