(4) (a) Using your hand manipulations, derive the 3-2-1 rotation matrix that relates the
aircraft's body-fixed velocity components $u, v, w$ to the aircraft's earth-fixed ve-
locity components $\dot{x}, \dot{y}, \dot{z}$, i.e., find the $L(\phi, \theta, \psi)$ matrix such that
$\begin{bmatrix} u \\ v \\ w \end{bmatrix} = L(\phi, \theta, \psi) \begin{bmatrix} \dot{x} \\ \dot{y} \\ \dot{z} \end{bmatrix}$.
(1)
(b) Find the inverse relation between the velocity components
$\begin{bmatrix} \dot{x} \\ \dot{y} \\ \dot{z} \end{bmatrix} = L^{-1}(\phi, \theta, \psi) \begin{bmatrix} u \\ v \\ w \end{bmatrix}$.
(2)
Note: $L$ is a rotation (orthonormal) matrix.
(c) Write a MATLAB code to obtain symbolic expressions of the matrices $L(\phi, \theta, \psi)$
and $L^{-1}(\phi, \theta, \psi)$ from the simple rotation matrices $L_1(\phi), L_2(\theta), L_3(\psi)$.
Note: Use the Matlab command \texttt{syms} to make symbolic manipulations.
