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Describing vectors in a different coordinate system

  1. Jul 30, 2014 #1
    The problem I am having is a problem in my textbook. It says that if we have xy Cartesian coordinate system, and if we then have a rotated coordinate system x'y', then to get the vector in the x'y' in terms of the xy system, we use the following arguments for the unit vectors:

    i' = icos[itex]\Phi[/itex] + jsin[itex]\Phi[/itex]

    j' = jcos[itex]\Phi[/itex] - isin[itex]\Phi[/itex]

    I don't understand how this was derived, or where it came from. I try to use the right-angle definition for trig ratios, but I keep getting different numbers, and don't see how this relation is true. I would realy appreciate it if somebody could provide a simple explanation.
  2. jcsd
  3. Jul 30, 2014 #2


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    Staff Emeritus
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    The derivation is mostly a matter of geometry. Perhaps this figure can clear things up:

  4. Jul 30, 2014 #3


    Staff: Mentor

    Wikipedia has an article on coordinate rotation


    midway down in the "two Dimensions" topic they show a matrix that transforms a vector from xy to x'y'

    In your case, I think you have the signs mixed up ie

    i' = i cos phi - j sin phi


    j' = i sin phi + j cos phi
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