Answer to Question #263934 in Complex Analysis for Kelly

Question

Express the following in rectangular and polar form, if

Z1 = 3+ 4i

Z2= 2+3i

1. Z1*Z2

2. Z1-Z2

3. Z1/Z2

4. |Z1|

5. |Z2|

(2)if Z1=50<30°and Z2=30<60°find in rectangular

form the following

1. |Z1|

2. |Z2|

3. |Z1|-|Z2|

4. Z1*Z2

5. |Z2-Z1|

6. |Z2|/|Z1|

Answer 1

(1)

1.

$$z_1\cdot z_2=(3+ 4i)(2+ 3i)=6+9i+8i-12$$

$$=-6+17i$$

$$|z_1\cdot z_2|=\sqrt{(-6)^2+17^2}=5\sqrt{17}$$

$$\tan \theta=\dfrac{17}{-6}, 90\degree<\theta<180\degree$$

$$z_1\cdot z_2=5\sqrt{17}\angle\big(180\degree-\tan^{-1}\dfrac{17}{6}\big)$$

2.

$$z_1-z_2=(3+ 4i)-(2+ 3i)=1+i$$

$$|z_1- z_2|=\sqrt{1^2+1^2}=\sqrt{2}$$

$$\tan \theta=\dfrac{1}{1}=1, 0\degree<\theta<90\degree$$

$$z_1-z_2=\sqrt{2}\angle45\degree$$

3.

$$z_1/ z_2=(3+ 4i)/(2+ 3i)=\dfrac{(3+4i)(2-3i)}{2^2+3^2}$$

$$=\dfrac{6-9i+8i+12}{13}=\dfrac{18}{13}-\dfrac{1}{13}i$$

$$|z_1/ z_2|=\sqrt{(\dfrac{18}{13})^2+(-\dfrac{1}{13})^2}=\dfrac{5\sqrt{13}}{13}$$

$$\tan \theta=\dfrac{-\dfrac{1}{13}}{\dfrac{18}{13}}=-\dfrac{1}{18}, 270\degree<\theta<360\degree$$

$$z_1/ z_2=\dfrac{5\sqrt{13}}{13}\angle\big(360\degree-\tan^{-1}\dfrac{1}{18}\big)$$

4.

$$|z_1|=\sqrt{3^2+4^2}=5$$

$$|z_1|=5+0i$$

$$|z_1|=5\angle0\degree$$

5.

$$|z_2|=\sqrt{2^2+3^2}=\sqrt{13}$$

$$|z_2|=\sqrt{13}+0i$$

$$|z_2|=\sqrt{13}\angle0\degree$$

(2)

1.

$$|z_1|=50+0i$$

2.

$$|z_2|=30+0i$$

3.

$$|z_1|-|z_2|=50-30=20$$

$$|z_1|-|z_2|=20+0i$$

4.

$$z_1\cdot z_2=50\cdot 30(\cos (30\degree+60\degree)+i\sin (30\degree+60\degree))$$

$$=1500(0+i(1))$$

$$z_1\cdot z_2=0+1500i$$

5.

$$z_1=50(\cos 30\degree+i\sin 30\degree)=25\sqrt{3}+25i$$

$$z_2=30(\cos 60\degree+i\sin 60\degree)=15+15\sqrt{3}i$$

$$z_2-z_1=(15-25\sqrt{3})+(15\sqrt{3}-25)i$$

$$|z_2-z_1|=\sqrt{(15-25\sqrt{3})^2+(15\sqrt{3}-25)^2}$$

$$=\sqrt{225-750\sqrt{3}+1875+675-750\sqrt{3}+625}$$

$$=10\sqrt{34-15\sqrt{3}}$$

$$|z_2-z_1|=10\sqrt{34-15\sqrt{3}}+0i$$

6.

$$\dfrac{|z_2|}{|z_1|}=\dfrac{30}{50}=\dfrac{3}{5}$$

$$|z_2|/|z_1|=\dfrac{3}{5}+0i$$