Answer to Question #347280 in Complex Analysis for Mapula Advice

Question

2. Given z1 = 2∠45o

, ; z2 = 3∠120o and z3 = 4∠180o

. Determine the following and leave your

answers in rectangular form:

(i)

(z1)² +z2

z2 +z3

(5)

(ii)

z1

z2z3

Answer 1

(i)

$$(z_1)^2=(2)^2\angle(2\cdot45\degree)=4i$$

$$z_2+z_3=3(-\dfrac{1}{2}+\dfrac{\sqrt{3}}{2}i)-4=-\dfrac{11}{2}-\dfrac{3\sqrt{3}}{2}i)$$

$$\dfrac{z_2}{z_2+z_3}=\dfrac{-3+3\sqrt{3}i}{-11-3\sqrt{3}i}$$

$$=\dfrac{(-3+3\sqrt{3}i)(-11-3\sqrt{3}i)}{121+27}$$

$$=\dfrac{33+9\sqrt{3}i-33\sqrt{3}i+27}{148}$$

$$=\dfrac{15}{37}-\dfrac{6\sqrt{3}}{37}i$$

$$(z_1)^2+\dfrac{z_2}{z_2+z_3}=4i+\dfrac{15}{37}-\dfrac{6\sqrt{3}}{37}i$$

$$=\dfrac{15}{37}+\dfrac{148-6\sqrt{3}}{37}i$$

(ii)

$$z_2z_3=3(4)\angle(120\degree+180\degree)=6-6\sqrt{3}i$$

$$\dfrac{z_1}{z_2z_3}=\dfrac{\sqrt{2}+\sqrt{2}i}{6-6\sqrt{3}i}$$

$$=\dfrac{(\sqrt{2}+\sqrt{2}i)(6+6\sqrt{3}i)}{36+108}$$

$$=\dfrac{6\sqrt{2}+6\sqrt{6}i+6\sqrt{2}i-6\sqrt{6}}{144}$$

$$=-\dfrac{\sqrt{6}-\sqrt{2}}{24}+\dfrac{\sqrt{6}+\sqrt{2}}{24}i$$