Complex numbers - I'm sure this is an easy - Argand diagram

[laura_a]

Homework Statement

OABC is a square on an Argand diagram. O Represents 0, A represents -4 + 2i, B Represents z, C represents w and D is the point where the diagonals of the square meet. (There are two possible squares that meet this criteria) Find the complex number represented by C and D in cartesian form.

Homework Equations

A represents -4 + 2i,

The Attempt at a Solution

I've worked out that AO is √20 and AC is √40 and the arg (AO) is 26°34'

I've also sketch the square on geogebra so I know the answers, just working on the negative side, C is -2 - 4i and D is -3 -1i (this is just from visual solution)

I've even started using the distance formula in A0 and AC and then simultaneous equations but it was far too messy considering how simple the answer is so I must just be missing one thing and I'm hoping for a kick start

Is there an easy way using my old maths skills or is there something in complex numbers that can help me? :) Maybe it is to do with the fact that A0 and C0 are perpendicular. Fun fact I just learned that in any square, the length of the diagonal is √2 times the length of a side... .mind blown - well I've just tested it in two made up squares and it worked both times... have yet to prove it by induction ;)

Thanks in advance

 

[andrewkirk]

Multiplying a complex number by $i$ rotates it anti-clockwise around O and by $-i$ rotates it clockwise. So if you multiply A by those two numbers you get the two possibilities for C.

Addition of complex numbers is a translation in the plane, so how can you work out B as a simple addition of two points you already have? When it comes to addition, you can treat complex numbers like vectors. That should enable you to find D in terms of two points you already have.

 

[laura_a]

I knew it was something simple, but the Fitzpatrick textbook is written for people who already know this stuff, not for people who are learning it so it's pretty tricky, thanks a heap!