Exploring the Argand diagram

$z_1$ and $z_2$ are complex numbers.
$z_3$ is the product $z_1*z_2$
Move the points $z_1$ and $z_2$ around to see what happens to $z_3$

Fix $z_1$, and move $z_2$ until $z_3$ is on the x-axis. What can you say about the trajectory of $z_2$ as you move it to keep $z_3$ on the x-axis?

Repeat the above for other values of $z_1$:

Can you make predictions about where $z_2$ needs to be for $z_3$ to be on the x-axis?
Can you predict where $z_2$ needs to be when you want $z_3$ to be at a given point on the x-axis?


Can you use your understanding of multiplication of complex numbers to explain how to make these predictions? Take a look at A Brief Introduction to Complex Numbers for a reminder of the notation and algebraic manipulation.

Now carry out the same process but this time aiming to keep $z_3$ on the y-axis.