Question

In this question the value of the Arguments of the complex numbers is in terms of radians, the value of π is already given all you need is the supply the fraction to make the response correct. 2. Answers should be given to two (2) significant figures. Given z=

2
​
+i and w=3i a) Find, (use fractions)
(i) ∣z∣
2
=
(ii) ∣w∣=
​

and Arg(z)=
and Arg(w)=
​

π
π
​
b) Determine simplified expressions for zw and
z
w
​
, giving the answers in the form r∠θ.

Answer 1

a)

(i) ∣z∣ = √(2² + 1²) = √5

(ii) ∣w∣ = 3

Arg(z) = tan⁻¹(1/2) = π/4

Arg(w) = π/2

b)

zw = (2 + i)(0 + 3i) = -3 + 6i

The magnitude of zw is √((-3)² + 6²) = 3√5.

The argument of zw is atan(6/-3) = -π/3 (since zw lies in the 3rd quadrant).

Then, zw = 3√5∠(-π/3)

z/w = (2+i)/(0+3i) = -i(2/3 - 1/3i)

The magnitude of z/w is √((2/3)² + (1/3)²) = √5/3.

The argument of z/w is atan(-1/2) = -π/4 (since z/w lies in the 4th quadrant).

Then, z/w = (√5/3) ∠(-π/4) = (1.50∠(-0.79))