Dot product between two vectors with complex components

Question

I have the following vectors:

$$\vec{a} = (-2i, 4, 5)$$

$$\vec{b} = (6, 5, i)$$

I need to get $<\vec{a}|\vec{b}>$. I did the normal procedure: $<\vec{a}|\vec{b}> = (-2i)(6) + (4)(5) + (5)(i) = 20 -7i$

But the answer is supposed to be $20+17i$, where did I had my mistake? Thanks.

Answer 1

Your mistake is that, when taking the dot product of vectors with complex components, you should take the complex conjugates of the components of one of the vectors.

The correct answer is $\overline{(-2i)}(6)+\overline{(4)}(5)+\overline{(5)}(i)=(2i)(6)+(4)(5)+(5)(i)$.