# Multiply two complex numbers

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?

• Multiplication of vectors can be defined in different ways. – Karl Aug 8 '15 at 13:36
• Do you mean Cross product for example? – Evgeny Semyonov Aug 8 '15 at 13:38
• @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). – Clement C. Aug 8 '15 at 13:38
• 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. – orion Aug 8 '15 at 13:40
• Short answer is that multiplication of complex numbers has nothing to do with vectors. – user21820 Aug 8 '15 at 13:58

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 $(\theta_a + \theta_b, ab)$.