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.

As J.W.Tanner explains, you have to take the complex conjugate of $a$. This preserves the useful property of the dot product that $<x;|;x>=0$ if and only if $x=0$. — TonyK, Jun 01 '20 at 18:01

J. W. Tanner · Answer 1 · 2020-06-01T17:55:39.673

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

edited Jun 01 '20 at 17:55

answered Jun 01 '20 at 17:50

J. W. Tanner

60,406

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I think "one of the vectors" should be "the first vector". – TonyK Jun 01 '20 at 17:56
@TonyK: I hedged because I think some have the convention to conjugate the components of the second vector, e.g., here – J. W. Tanner Jun 01 '20 at 17:58
You may be right, but I've never seen that. – TonyK Jun 01 '20 at 17:59
@TonyK: cf. discussion here – J. W. Tanner Jun 01 '20 at 18:06
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I stand corrected! – TonyK Jun 01 '20 at 18:18

Dot product between two vectors with complex components

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