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Complex Vectors rotation

  1. Jul 12, 2009 #1
    1. The problem statement, all variables and given/known data
    Can someone explain to me why complex vectors of the following form will rotate counterclockwise in the x-y plane [with a velocity of w],

    [tex]\hat{v}(t) = cos(\omega t)\hat{x} + sin(\omega t)\hat{y}[/tex] (#1)

    And why the following equation, the unit vector rotates in the clockwise direction in the x-z plane,

    [tex]\hat{v}(t) = -sin(\omega t)\hat{x} + cos(\omega t)\hat{z}[/tex] (#2)


    2. An attempt:
    So I approached this problem by assuming [tex]\omega = 1; t = 0, t = \frac{\pi}{2}, t= \pi, t = \frac{3\pi}{2}, t = 2\pi[/tex]. When I computed this values, it seemed to me [tex]\hat{v}(t)[/tex] increased in the counterclockwise direction for both equations (#1) and (#2). Can someone explain to me the nature of the rotation for complex vectors?

    thanks,


    JL
     
  2. jcsd
  3. Jul 12, 2009 #2

    D H

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    Where are the complex numbers in this problem? It appears you are talking about rotation in real 3-space.

    What do you mean by clockwise and counterclockwise? Imagine a transparent clock, one whose hands are visible from behind. When viewed from the back the hands will appear to be moving counterclockwise.

    The right hand rotation rule provides a much better basis for terminology. In the first problem, the angular velocity vector points along +z axis. Where does it point in the second problem?
     
  4. Jul 12, 2009 #3
    I think these "Complex Vectors" is in relation to "Phasors". When i say the orientation, for example clockwise, I mean the unit vector [tex]\hat{v}(t)[/tex] is rotating about a chosen plane [for instance x-y plane] in that respective direction. As it rotates, [since it has a fixed magnitude], it will trace a circle as the angular velocty [tex]\omega[/tex] increases. I hope that helps.
     
  5. Jul 12, 2009 #4

    D H

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    You don't need to help me. You are the one asking the question. This looks like homework, so I cannot by the rules of this site tell you the answer straight out. Besides, the question is ill-formed. What do you mean by clockwise? Think of the transparent clock.
     
  6. Jul 12, 2009 #5
    If I cannot help you understand my question, how will I get my question answered? I am simply typing up definitions from my notes and asking for an interpretation from a different source- and right now that is you. And to my knowledge, the complex vector will rotate in the clockwise direction, by the equation (#1) of above. I am trying to make sense of my notes, and I am having difficulty right now, sorry for conjuring up bad questions, as I've never taken any EE courses before but am interested.

    Thanks,


    JL
     
    Last edited: Jul 12, 2009
  7. Jul 12, 2009 #6

    D H

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    It isn't a complex vector. You have no complex numbers.

    At t=0, the vector in the first equation will be oriented along the plus-x axis. A short time later, the vector's x component won't have changed much and the y component will be small but positive. The reason this first equation is said to rotate clockwise is because the x and y axes are typically represented as being horizontal and vertical, respectively, and with increasingly positive values to the right and upward. If you drew the +x axis to the left and the +y axis down, you would get the opposite picture.

    The second problem: What made you use the x-z plane, as opposed to the z-x plane? What direction are the axes oriented?
     
  8. Jul 12, 2009 #7

    Redbelly98

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    (1) is reasonably unambiguous, because there is a preferred convention for viewing the x-y plane (+x is to the right, +y is upward). You can think of this as being viewed from the positive z side of the x-y plane.

    However, for (2), it depends on what direction the x-z plane is to be viewed from: positive y or negative y side of the plane?

    Is there a figure with the problem statement that would clarify this?
     
  9. Jul 12, 2009 #8
    It says "Write the equation of a unit vector that rotates clockwise in the x-z plane when viewed from the positive y-axis. The vector should poin in the z direction at t = 0. Then couple lines below, it tells me,
    "In order that it rotates clockwise: [tex]
    \hat{v}(t) = -sin(\omega t)\hat{x} + cos(\omega t)\hat{z}
    [/tex]."
     
  10. Jul 12, 2009 #9

    D H

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    There you go. Right hand rotation rule, which you can google. If you don't understand what you find, ask away.
     
  11. Jul 12, 2009 #10
    I think I knew the right hand rule- or hope so, but I am trying to think of why the equation itself causes the vector to rotate. So I let wt = 0, and thus z = 1, when I let wt = pi/2 then x = -1. If I continue on, it seems to me as values inside the equation increases, a rotation is clockwise. Is that a reasonable way of thinking? And can I apply this way of thinking to the other equation that I asked about?

    thanks,


    Jeff
     
  12. Jul 12, 2009 #11

    Redbelly98

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    Yes, that's a good way to look at it. ωt = 0, π/2, π, and 3π/2. It should be clear by the time you get to π or even π/2.
     
  13. Jul 12, 2009 #12
    Oh cool, thanks a lot. I actually was thinking in this way earlier, but as I was doing it, I was picturing the x-axis as the opposite orientation as it should be, and that just threw me in a loop haha. Thanks for the help guys.
     
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