$(20$ pt$)$ See Figure $4.$ This system consists of three rotating sticks, all attached at their ends. The leftmost side is fixed at $O.$ There are four frames, each of which consists of two unit vectors: $\{I$:vec$(c{)}_x,$vec$(c{)}_y\},\{B_1$:vec$(c{)}_{r1},$vec$(c{)}_1\},\{B_2$:vec$(c{)}_{r2},$vec$(c{)}_2\},$ and $\{B_3$:vec$(c{)}_{r3},$vec$(c{)}_3\}.$ The unit vectors for $B_1,B_2,$ and $B_3$ rotate with a designated frame. Compute the unit vectors do not define their positions; in other words, you may consider that they can be moved spatially and only take their directions. All time derivatives should be given with respect to Frame I. Use ${}_1^{},{}_2^{},$ and ${}_3^{}$ for designated angular speeds.

Figure $4$: Multi$-$body system for Problem $5.$