# Definition: Dot Product of Complex Numbers

Let $\begin{array}{rcl} x & := & a + b i\\ y & := & c + d i \end{array}$ be two complex numbers, i.e. vectors represented by linear combinations of the basis $$\{1,i\}$$. We define the dot product of these two complex numbers as follows

$\begin{array}{rcl} \langle x,y\rangle &:=&\Re(x\cdot y^*)\\ &=&\Re((a+bi)\cdot (c-di))\\ &=&\Re(ac-adi+bci-bdi^2)\\ &=&\Re(ac-bd\cdot(-1)+(bc-ad)i)\\ &=&\Re((ac+bd)+(bc-ad)i)\\ &=&ac+bd \end{array}$

where $$y^*$$ is the complex conjugate of $$y$$ and $$\Re(x\cdot y^*)$$ denotes the real part of the complex number $$x\cdot y^*$$. Please note that the dot product of complex numbers is always a real number. The following interactive figure shows two complex numbers (blue), which you can drag to see how their position influences the dot product (red).

Fun questions: * When is the dot product negative? * When is the dot product positive? * When does the dot product equal zero?

Definitions: 1

Github: ### References

#### Bibliography

1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013