Answer
$$\left( {\bf{a}} \right) - \sqrt 3 - i,\,\,\left( {\bf{b}} \right)\,\,2\left( {\cos 210^\circ + i\sin 210^\circ } \right)$$
Work Step by Step
$$\eqalign{
& {\text{From the graph we can see that the coordinates of the vector are}} \cr
& {\text{ }}x = - \sqrt 3 \,{\text{and }}y = - 1 \cr
& \cr
& \left( {\text{a}} \right){\text{ Its rectangular form is}} \cr
& z = x + yi \cr
& z = - \sqrt 3 - i \cr
& \cr
& \left( {\text{b}} \right){\text{ Its trigonometric }}\left( {{\text{polar}}} \right){\text{ form}} \cr
& z = - \sqrt 3 - i \cr
& {\text{Use }}r = \sqrt {{y^2} + {y^2}} {\text{ and }}\theta = {\tan ^{ - 1}}\left( {\frac{y}{x}} \right),{\text{ so}} \cr
& r = \sqrt {{{\left( { - \sqrt 3 } \right)}^2} + {{\left( { - 1} \right)}^2}} = \sqrt 4 = 2 \cr
& \theta = {\tan ^{ - 1}}\left( {\frac{{ - 1}}{{ - \sqrt 3 }}} \right) + 180 \cr
& \theta = 210^\circ \cr
& {\text{write the vector in the trigonometric form }}r\left( {\cos \theta + i\sin \theta } \right) \cr
& = 2\left( {\cos 210^\circ + i\sin 210^\circ } \right) \cr} $$
