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 \(P_2\).
Complex Numbers with Vector Addition
(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 \( z_0 + z_1 + z_2 +z_3 + z_4 = 0 \).

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