# Complex Numbers with Vector Addition

Complex Numbers with Vector Addition is obtained using the sides of triangles, the sum of sides of two sides of a triangle is greater than the other side.

### Worked Examples of Complex Numbers with Vector Addition

Suppose $\displaystyle 0 \lt \alpha, \ \beta \lt \frac{\pi}{2}$ and define complex numbers $z_n$ by
$$z_n = \cos(\alpha + n \beta) + i \sin(\alpha + n \beta)$$
for $n = 0, \ 1, \ 2, \ 3, \ 4$. the points $P_0, \ P_1, \ P_2$ and $P_3$ are the points in the Argand diagram that correspond to the complex numbers $z_0, \ z_0+z_1, \ z_0+z_1+z_2$ and $z_0+z_1+z_2+z_3$ respectively. The angles $\theta_0, \ \theta_1$ and $\theta_2$ are the external angles at $P_0, \ P_1$ and $P_2$.

(a)    Using vector addition, explain why $\theta_0 = \theta_1 = \theta_2 = \beta$.

(b)    Show that $\angle P_0O P_1 = \angle P_0 P_2 P_1$.

(c)    Explain why $O P_0 P_1 P_2$ is a cyclic quadrilateral.

(d)    Show that $P_0 P_1 P_2 P_3$ is a cyclic quadrilateral.

(e)    Explain why $O P_0 P_1 P_2 P_3$ are concyclic.

(f)    Find the value of $\beta$ if $z_0 + z_1 + z_2 +z_3 + z_4 = 0$.