### Home > PC3 > Chapter 12 > Lesson 12.3.1 > Problem12-133

12-133.

Let $z_{1} = 1 − i$ and $z_2 = −1 + (\sqrt { 3 })i$.

1. What is the distance between $z_{1}$ and $z_{2}$?

$\sqrt{(1-(-1))^2+(-1-\sqrt3)^2}$

2. What is the midpoint of $z_{1}$ and $z_{2}$?

$\frac{(1-i)+(-1+\sqrt3)i}{2}$

3. What are the modulus and argument of $z_{2}$?

Graph the point associated with$z_{2}$.
How for is the point from the origin? This is the modulus.
What angle does the point make with the positive $x$-axis? This is the argument.

4. Evaluate $(z_2)^5$.

De Moivre’s Theorem: Given the complex numbers $z=r(\cos(\theta)+i\sin(\theta))$, when $z$ is raised by a positive integer $n$, the result is: $z ^ { n } = [ r ( \operatorname { cos } ( \theta ) + i \operatorname { sin } ( \theta ) ) ] ^ { n }$ or $z^n=r^n(\cos(n\theta)+i\sin(n\theta))$