In school, I was taught that complex numbers could be represented using vectors. My teacher used this technique on sample problems. It had never failed me and I've been able to use it to solve many problems involving complex numbers. However, today I realized I didn't understand what I was doing.
Consider a complex number $z = 3 + 4i.$ Let $M$ be the point representing $z$ on the complex plane $Oxy.$ I was taught that $|z| = |\vec{OM}| = \sqrt{3^2 + 4^2} = 5.$ This is certainly true for $z,$ and I could visualize why this is also true in terms of vector $\vec{OM}$.
Likewise, $|z^2| = |\vec{OM}\cdot\vec{OM}| = OM^2 = 3^2 + 4^2 = 25.$
Let $N$ represent the complex number $w.$ We also did operations like:
$|z – w| = |\vec{OM} – \vec{ON}| \implies |z – w|^2 = |\vec{OM} – \vec{ON}|^2 = OM^2 + ON^2 – 2\vec{OM}\cdot\vec{ON} $
This way, suppose we know the values of $|z – w|, OM, ON,$ we could find the value of $\vec{OM}\cdot\vec{ON}.$
However, today I just found that doing this could lead to weird results. Consider two complex numbers $a = 1$ and $b = i,$ represented by $A$ and $B$ on the complex plane $Oxy.$ Then:
$|a \cdot b| = |1 \cdot i| = 1$
However, $|\vec{OA}\cdot\vec{OB}| = 0$
So obviously, $|a \cdot b| \ne |\vec{OA}\cdot\vec{OB}|$
Given my inability to explain this, I clearly have not understood the true nature of what I have been doing. I hope to receive some advice on this.
My questions: What makes the vector representations work in all the first examples but not in the last example? To what extent could we use vector representation to produce an operation equivalent to one in complex number algebra?
Thank you very much!
Best Answer
The heart of your issue is this: the dot product of two complex numbers and the usual product of complex numbers is not the same product. Full stop. There is no "in some cases it's the same and in others it isn't"*. If you are ever exchanging a dot product for a normal product or vice versa, you are most likely making a mistake.
The only algebraic operation on the complex numbers that corresponds to a geometric operation on vectors is the sum of complex numbers, which corresponds to the vector sum.
* Technically speaking, it is the same for the boring case where all numbers involved are purely real. But there's no need to bring vector geometry into it then, because then everything happens on a 1d line anyway.