(Solved): 2. Two aircraft are flying side-by-side. Body coordinate frame A is centered on the left aircraft, ...

2. Two aircraft are flying side-by-side. Body coordinate frame $\mathrm{A}$ is centered on the left aircraft, and body coordinate frame $\mathrm{B}$ is centered on the right aircraft. The coordinate frames have roll, pitch, and yaw axes defined by $A:\left\{{\mathbf{i}}^A,{\mathbf{j}}^A,{\mathbf{k}}^A\right\}$ and $B:\left\{{\mathbf{i}}^B,{\mathbf{j}}^B,{\mathbf{k}}^B\right\}$. The coordinate frames are initially aligned. Aircraft A observes a third aircraft, $\mathrm{C}$, in the distance, and moves to intercept it at $t=0$. The constant 3-2-1 Euler angle rates of coordinate frame A relative to coordinate frame B are $\dot{\varphi }=-0.2,\dot{\theta }=0.3$, and $\dot{\psi }=-0.4$, all in $\mathrm{rad}/\mathrm{s}$. (a) What is ${\mathbf{\omega }}_A^B$, the angular velocity vector of frame A relative to frame B, expressed in A frame components, at $t=2$ seconds? (b) While in pursuit, aircraft A observes aircraft C at position ${\mathbf{r}}_A^A=(3,2,4{)}^T\mathrm{km}$ at $t=2$ seconds. The time derivative of this position vector relative to $\mathrm{A}$ is $\frac{d}{dt}{\mathbf{r}}_A^A=(-70,-50,-100{)}^T$ $\mathrm{m}/\mathrm{s}$. What is the time derivative of the same position vector relative to aircraft $\mathrm{B},\frac{d}{dt}{\mathbf{r}}_A^B$ ?