Why isn't the imaginary component j included in the complex vector equation?

[jeff1evesque]

Problem/Statement
The complex vector, $$\hat{v}(t) = cos(\omega t)\hat{x} + sin(\omega t)\hat{y}$$ is the unit vector $$\hat{v}(t)$$ expressed in instantaneous form.

Question
What I am wondering is, why is there no imaginary component "j" in say the sin component for the equation above?

Can we express a general notation for complex vectors as,
$$\hat{v}(t) = [cos(\omega t) + sin(\omega t)]\hat{x}] + [cos(\omega t) + sin(\omega t)]\hat{y}]$$? Shouldn't that be the notation for the instantaneous form also?Thanks,Jeff

 

[Tac-Tics]

Weird. You'd have to consult the source of the question (either your book or your professor or whatnot).

With the hats over the x and y, it almost looks like they are mixing R^2 and C. The two are, in fact, isomorphic, and anything you can say about complex numbers translates simply to a statement about vectors in a 2D plane.

If it were me, I'd make the assumption that they meant x = 1 and y = j. That way, v(t) is a complex number. (In fact, v would be the exponential function, $$e^{it}$$).

 

[jeff1evesque]

Weird. You'd have to consult the source of the question (either your book or your professor or whatnot).

With the hats over the x and y, it almost looks like they are mixing R^2 and C. The two are, in fact, isomorphic, and anything you can say about complex numbers translates simply to a statement about vectors in a 2D plane.

If it were me, I'd make the assumption that they meant x = 1 and y = j. That way, v(t) is a complex number. (In fact, v would be the exponential function, $$e^{it}$$).

Yea, I'm not sure. I think I will ask the teacher tomorrow. Oh I think I meant the following (as well),

Can we express a general notation for complex vectors as,
$$\hat{v}(t) = [[cos(\omega t) + sin(\omega t)]\hat{x} + j[cos(\omega t) + sin(\omega t)]\hat{y}]$$? Shouldn't that be the notation for the instantaneous form also?