$(25$ points$)$ Consider a coordinate system given by

hat$(i{)}_1=$hat$(i)$

hat$(j{)}_1=$hat$(j)$

hat$(k{)}_1=$hat$(k)$

If this coordinate system is rotated by $=70$ about the hat$(k{)}_1$ axis, $=15$ about the hat$(j{)}_1$ axis and $=50$ about the hat$(i{)}_1$ axis, determine the new coordinate axes hat$(i{)}_2,$hat$(j{)}_2,$ and hat$(k{)}_2$

$2.(15$ points$)$ Consider the following non$-$linear relation

$h^{}=$Usin$$

Linearize the equations about the steady$-$state values $h_0,U_0,$ and ${}_0,$ with the small perturbations taken as $h,U,$ and $.$

$3.(10$ points$)$ Consider a hypothetical aircraft whose mass is $1250$ kg and wing area $10m^2.$ If the slope of the $C_m$ versus $C_L$ plot is $-0\mathrm{.}4,$ and the pitching moment at zero lift is equal to $0\mathrm{.}2,$ determine the trim lift coefficient. If the aircraft was flying at an altitude where the freestream density is $0\mathrm{.}5k\frac{g}{m^3},$ determine the velocity of the aircraft.

$4.(40$ points$)$ Consider an aircraft with the following parameters

$\backslash$table$[[$b$,10$ m$],[$S$,20m^2$