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?
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