An expert .golf player hits a ball from the tree to the hole in $4$ shots. The first shot is $200$ m at $20\mathrm{.}\mathrm{⁰}$ north of east, the second is $100$ m at $45\mathrm{.}\mathrm{⁰}$ south of east, the third is $55\mathrm{.}0$ m at $60\mathrm{.}\mathrm{⁰}$ west of north and finally the fourth is $10\mathrm{.}0$ m at $30\mathrm{.}\mathrm{⁰}$ west of south.

 a$)$ Express the four moves in polar coordinates.

 b$)$ Express the four moves in Cartesian coordinates.

c$)$ Express the sum of the vector in terms of the unit vectors.

d$)$ If the position of the hole is $(205$ m$,-5\mathrm{.}00$ $)$ from the tree box, determine if the ball fell into the hole. $\mathrm{}$