What is the significance of complex vectors in electromagnetism?

[tomizzo]

I've recently began a course on electromagnetism and have started dealing with complex vectors. I have a couple questions to ask:

Regarding the general concept of complex vectors, I am curious what these actually represent. Refer to attached equation. Am I correct to believe that this equation represents a wave as a function of time traveling through 3D space? And even though there are 3 different sinusoidal functions, they represent a single propagating signal?

Now moving onto the mathematics, I have a question regarding an example problem. Refer the second equation attached. This equation is a phasor representation of a wave in 2D space. I am having trouble translating this phasor representation into the time domain. I understand how the x component translates into cos(wt) but I am having trouble in understanding how the y component translates. Specifically, I don't understand why it is negative...

 

[AlephZero]

$A(t)$ is the real part of $(\hat x + j\hat y)e^{j\omega t}$.

$e^{j\omega t} = \cos \omega t + j \sin \omega t$.

And $j^2 = -1$.

 

[tomizzo]

$A(t)$ is the real part of $(\hat x + j\hat y)e^{j\omega t}$.

$e^{j\omega t} = \cos \omega t + j \sin \omega t$.

And $j^2 = -1$.

Ahh, thank you AlephZero. I apparently at times forget about the common frequency within the two components.

Regarding my first question, any insight on that?

 

[AlephZero]

Regarding my first question, any insight on that?

Sorry, but without any context it's just an equation with a lot of undefined variables in it.

Apparently three different things (or three components of the same thing?) are varying at the same frequency but with different amplitudes and phases, and for some reason they are added together. I think more than that is just guessing.

 

[Babadag]

Actually, what is the meaning of "vector" it is a physical real vector-as force or magnetic field density, electric field or magnetic field and other. What we call vector, usually, it is only the complex [symbolic] representation of alternative sinusoidal variation of current or voltage. However there are also "complex vectors" actual vectors in complex representation. See:
http://www.ismolindell.com/publications/monographs/pdf/Methchap1.pdf