A sailboat race course consists of four legs, defined by the displacement vectors vec$(A),$vec$(B),$vec$(C),$ and vec$(D),$ as the drawing indicates. The magnitudes of the first three vectors are $A=3\mathrm{.}10km,B=5\mathrm{.}10km,$ and $C=5\mathrm{.}30km.$ The finish line of the course coincides with the starting line. Using the data in the drawing, find $($a$)$ the distance of the fourth leg and $($b$)$ the angle $.$