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multiplication of two complex numbers - it's the same as multiplication of vectors.

From physics i know that's result of multiplication of two vectors - it's a number.

But when we multiply complex numbers - it's a vector. It's some kind of contradiction, isn't it?

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    $\begingroup$ Multiplication of vectors can be defined in different ways. $\endgroup$ – Karl Aug 8 '15 at 13:36
  • $\begingroup$ Do you mean Cross product for example? $\endgroup$ – Evgeny Semyonov Aug 8 '15 at 13:38
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    $\begingroup$ @OP: you may be confusing the notion of multiplication of vectors (which give a vector, and can be defined in several ways), and dot product (which gives a scalar/number). $\endgroup$ – Clement C. Aug 8 '15 at 13:38
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    $\begingroup$ This is a strange thing to say... a really confusing comparison. Unless this is in the context of Clifford algebra, it doesn't make much sense. $\endgroup$ – orion Aug 8 '15 at 13:40
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    $\begingroup$ Short answer is that multiplication of complex numbers has nothing to do with vectors. $\endgroup$ – user21820 Aug 8 '15 at 13:58
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If the complex numbers are written with real and imaginary parts, $a+ib$ and $c+id$, then: $$(a+ib)(c+id) = ac - bd + i(ad + bc)$$

If the complex numbers are written as a direction and a magnitude, $(\theta_a, a)$ and $(\theta_b, b)$, then their product is the complex number $(\theta_a + \theta_b, ab)$.

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