Problem $1(10$ pts$)$

Refer to Fig. $1$: a telescopic antenna has three degrees of freedom, $,,$ and $,$ i$.$e$.,$ the user can change two angles $($ and $),$ and one length $().$

$($a$)$ Determine the angular velocity between $T\{hat(t{)}_1,$hat$(t{)}_2,$hat$(t{)}_3\}$ and $B\{hat(b{)}_1,$hat$(b{)}_2,$hat$(b{)}_3\},$ i$.$e$.,{}^{tb}.$ Express it in terms of pts$)$

$($b$)$ Determine the angular velocity between $B\{hat(b{)}_1,$hat$(b{)}_2,$hat$(b{)}_3\}$ and $E\{hat(e{)}_1,$hat$(e{)}_2,$hat$(e{)}_3\},$ i$.$e$.,{}^{be}.$ Express it in terms of $E\{hat(e{)}_1,$hat$(e{)}_2,$hat$(e{)}_3\}.(2\mathrm{.}5$ pts$)$

$($c$)$ Determine the angular velocity between $T\{hat(t{)}_1,$hat$(t{)}_2,$hat$(t{)}_3\}$ and $E\{hat(e{)}_1,$hat$(e{)}_2,$hat$(e{)}_3\},$ i$.$e$.,{}^{te}.$ Express it in terms of $E\{hat(e{)}_1,$hat$(e{)}_2,$hat$(e{)}_3\}.(2\mathrm{.}5$ pts$)$

$($d$)$ Determine the velocity of $P$ with respect to $E\{hat(e{)}_1,$hat$(e{)}_2,$hat$(e{)}_3\}.$ Express it in terms of $E\{hat(e{)}_1,$hat$(e{)}_2,$hat$(e{)}_3\}.(2\mathrm{.}5$ pts$)$

Fig. $1$: $E\{hat(e{)}_1,$hat$(e{)}_2,$hat$(e{)}_3\}$ : inertial frame. $B\{hat(b{)}_1,$hat$(b{)}_2,$hat$(b{)}_3\}$ : base frame. $T\{hat(t{)}_1,$hat$(t{)}_2,$hat$(t{)}_3\}$ : telescopic antenna frame. Note that hat$(b{)}_1$ and hat$(b{)}_2$ are in the same plane as hat$(e{)}_1$ and hat$(e{)}_2.$ Similarly, hat$(t{)}_1$ and hat$(t{)}_3$ are in the same plane as hat$(b{)}_1$ and hat$(b{)}_3.$