Answer
$$\left( {\bf{a}} \right)2 + 2i,\,\,\left( {\bf{b}} \right)\,\,2\sqrt 2 \left( {\cos 45^\circ + i\sin 45^\circ } \right)$$
Work Step by Step
$$\eqalign{
& {\text{From the graph we can see that the coordinates of the vector are}} \cr
& {\text{ }}x = 2\,{\text{and }}y = 2 \cr
& \cr
& \left( {\text{a}} \right){\text{ Its rectangular form is}} \cr
& z = x + yi \cr
& z = 2 + 2i \cr
& \cr
& \left( {\text{b}} \right){\text{ Its trigonometric }}\left( {{\text{polar}}} \right){\text{ form}} \cr
& z = 2 + 2i \cr
& {\text{Use }}r = \sqrt {{x^2} + {y^2}} {\text{ and }}\theta = {\tan ^{ - 1}}\left( {\frac{y}{x}} \right),{\text{ so}} \cr
& r = \sqrt {{{\left( 2 \right)}^2} + {{\left( 2 \right)}^2}} = 2\sqrt 2 \cr
& \theta = {\tan ^{ - 1}}\left( {\frac{2}{2}} \right) \cr
& \theta = 45^\circ \cr
& {\text{write the vector in the trigonometric form }}r\left( {\cos \theta + i\sin \theta } \right) \cr
& = 2\sqrt 2 \left( {\cos 45^\circ + i\sin 45^\circ } \right) \cr
& \cr
& \left( {\bf{a}} \right)2 + 2i,\,\,\left( {\bf{b}} \right)\,\,2\sqrt 2 \left( {\cos 45^\circ + i\sin 45^\circ } \right) \cr} $$
