$($a$)$ Indicate the values for $,,$ and $.$

$=12$

$=04$

$=05$

$($b$)$ Write the matrix product that gives the resulting rotation matrix if first you rotate $$ about the current $y-$axis and then you rotate $$ about the current $x-$axis. You will need to include three matrices: $R_{y,},R_{x,},$ and the corresponding transformation matrix.

$($c$)$ If the target object also has a displacement of $7$ units in the $z_0$ direction, find the corresponding homogenous vector $d$ and the homogenous transformation matrix $H.$

$($d$)$ If a sensor is located at $p^1=[{]}^T,$ find the position measured from frame $0(p^0).$

$($e$)$ Synthesize the inverse of the homogenous transformation matrix $H^{-1}$ and show the procedures that the reverse transformation can lead to $p^1.$