Q$2$: $(10$ marks$)$

Knowing that the total transformation of a spherical coordinate system is

$T_{sph}(r,,)=\left[\begin{array}{cccc}CC& -S& SC& rSC\\ CS& C& SS& rSS\\ -S& 0& C& rC\\ 0& 0& 0& 1\end{array}\right]$

the hand orientation of the frame is later restored in order to be parallel to the reference frame, and the matrix representing it is described as:

$T_{}=\left[\begin{array}{cccc}1& 0& 0& 3\mathrm{.}1375\\ 0& 1& 0& 2\mathrm{.}195\\ 0& 0& 1& 3\mathrm{.}214\\ 0& 0& 0& 1\end{array}\right]$

a$.$ Find the necessary values of $r,,$ to achieve this location.

b$.$ Find the components of the original matrix $n,0,$a vectors for the hand before the orientation was restored.

$3$