Question

Let z1 = 2 − 2i and z2 = (1 − i) + √3(1 + i).

a. What is the modulus and argument of z1?
b. Write z1 in polar form. Explain why the polar and rectangular forms of a given complex number
represent the same number.
c. Find a complex number ????????, writen in the form ???????? = a+ ii, such that ????????z1 = z2.
d. What is the modulus and argument of ?????????
e. Write ???????? in polar form.
f. When the points z1 and z2 are ploted in the complex plane, explain why the angle betwen z1 and
z2 measures arg(????????)

Answer 1

Explanation:

z₁ = 2 − 2i

z₂ = (1 − i) + √3(1 + i) = (1 + √3) + (√3 - 1) i

a) We get the modulus of z₁ as follows

║z₁║ = √((2)²+(-2)²) = 2

now we find the argument

α = Arctan (-2/2) = Arctan (-1) = -45º ⇒ α = 360º + (-45º) = 315º

b) z₁ = 2 Cis 315º

Although the complex number is in binomic or polar form, its representation must be the same, since the complex number is the same, only that it is expressed in two different forms. The modulus represents the distance from the origin to the point. The degree of rotation is the angle from the x-axis. When the polar form is expanded, the result is the rectangular form of a complex number.

c) If z₀*z₁ = z₂ and z₀ = a + b i

we have

(a + b i)*(2 − 2i) = (1 + √3) + (√3 - 1) i

⇒ 2a + 2bi - 2ai - 2bi² = (1 + √3) + (√3 - 1) i

⇒ 2a + 2bi - 2ai - 2b(-1) = (1 + √3) + (√3 - 1) i

⇒ 2a + 2b + 2bi - 2ai = (1 + √3) + (√3 - 1) i

⇒ 2 (a + b) + 2 (b - a) i = (1 + √3) + (√3 - 1) i

Now we can apply

2 (a + b) = 1 + √3

2 (b - a) = √3 - 1

Solving the system we get

a = 1/2

b = √3 / 2

Finally

z₀ = (1/2) + (√3 / 2) i

d) ║z₀║ = √((1/2)²+(√3 / 2)²) = 1

α = Arctan ((√3 / 2)/(1/2)) = 60º

e) z₀ = Cis 60º

f) Since z₂ = z₀*z₁, then z₂ is the transformation of z₁ rotated counterclockwise by arg(w) which is 60º