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I am reading about phasors. Everywhere it says that phasors are complex numbers and vectors which is obvious given that every complex number is a two-dimensional vector. But there are a lot of circumstances where you have to multiply two phasors, which is done regarding them as complex numbers and so we do

$$ {\bf A}\cdot {\bf B} = (A\angle{\phi_{A}})\cdot (B\angle{\phi_{B}}) = AB\angle(\phi_{A}+\phi_{B}) $$

  1. Why multiplying two phasors is done like that and not as a dot product?
  2. Why we choose one way of multiplication (the one used for complex numbers) and not the other (the one used in vector algebra)?
  3. As phasors are regarded as complex numbers and vectors when do we use complex numbers' qualities and where vectors' ones?
  4. When we multiply them like that are they still regarded as vectors? If yes then why can't I find any mention of "complex type-multiplication" for vectors?

In general this question essentially is, as complex numbers are considered as two-dimensional vectors when do we use their vector nature?

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