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

$$\mathrm { A } ( - 3 ; 1 ; 3 ) , \mathrm { B } ( 2 ; 2 ; 3 ) , \mathrm { C } ( 1 ; 7 ; - 1 ) , \mathrm { D } ( - 4 ; 6 ; - 1 ) \text { and K(-3;14;14). }$$

\begin{enumerate}
  \item a. Calculate the coordinates of the vectors $\overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { DC } }$ and $\overrightarrow { \mathrm { AD } }$.\\
b. Show that the quadrilateral ABCD is a rectangle.\\
c. Calculate the area of rectangle ABCD.
  \item a. Justify that the points $\mathrm { A } , \mathrm { B }$ and D define a plane.\\
b. Show that the vector $\vec { n } ( - 2 ; 10 ; 13 )$ is a normal vector to the plane (ABD).\\
c. Deduce a Cartesian equation of the plane (ABD).
  \item a. Give a parametric representation of the line $\Delta$ orthogonal to the plane (ABD) and passing through point K.\\
b. Determine the coordinates of point I, the orthogonal projection of point K onto the plane (ABD).\\
c. Show that the height of the pyramid KABCD with base ABCD and apex K equals $\sqrt { 273 }$.
  \item Calculate the volume $V$ of the pyramid KABCD.
\end{enumerate}

Recall that the volume V of a pyramid is given by the formula:

$$V = \frac { 1 } { 3 } \times \text { base area × height. }$$