# Simple complex number question

$$z_1z_2 = -1 + 2i$$
$$\frac{z_1}{z_2} = \frac{11}{5} + \frac{2}{5}i$$

Given that the origin, z1z2, z1/z2 and z3 are vertices of a rhombus, find z3.

I've drawn a sketch on a Argand diagram and the sketch is fine, but to find z3, they have done z1z2 + z1/z2 , but would this not give you a diagonal which joins z1z2 and z1/z2 instead of z3? How would it give you z3, if anyone could explain

vela
Staff Emeritus
Homework Helper
No, it wouldn't give you the diagonal joining ##z_1z_2## and ##z_1/z_2##.

Consider a simpler case where the vertices are at (0, 0), (1, 0), and (0, i). Where would the other corner of the square lie and how are its coordinates related to (1, 0) and (0, i)?

No, it wouldn't give you the diagonal joining ##z_1z_2## and ##z_1/z_2##.

Consider a simpler case where the vertices are at (0, 0), (1, 0), and (0, i). Where would the other corner of the square lie and how are its coordinates related to (1, 0) and (0, i)?

I see.

Does this only work on Argand diagrams and quadrilaterals?

I would think of the two given points as vectors. If z1z2 is one side, and z1/z2 is another, then their sum gives the position vector of the opposite corner.