I am currently studying linear algebra, specifically the orthogonality of a set of vectors. I learnt that the inner product of the 2 vectors in the field $mathbb {C}$ can be expressed by the sum of multiplication of the one’s counterpart with another’s **conjugated** one.

$$ <u,v> = sum a_i.bar bi $$

This formula can apply to a set of $mathbb {R}$ too. My question is why in the field $mathbb{C}$ we need to use the **conjugated** vector instead of the normal one.

Thank you in advance