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Coordinate transformations Spherical to Cartesian

  1. Jan 6, 2009 #1
    I would like to transform a vector from Spherical to cartesian coordinate system. But the question is probably not that straight forward. :(

    I have a vector say [tex]E = E_r~\hat{r}+E_{\theta}~\hat{\theta}+E_{\phi}~\hat{\phi}[/tex].

    But I know only the cartesian coordinate from where it starts, say [tex](x,y,z)[/tex] and I do not know where it ends. So I am unable to find angles [tex]\theta[/tex] and [tex]\phi[/tex] for computing the transformation matrix [tex]R[/tex] that transforms the vector [tex]E[/tex] to cartesian system. This [tex]R[/tex] is the usual matrix with sines and cosines of [tex]\theta[/tex] and [tex]\phi[/tex] and can be seen here.

    So how do I go about it. Is there even a way to do this. Once again this is not a homework question and is for a small project that I am doing. There aren't any homeworks at this time of the year. :)

    Appreciate any form of help.

  2. jcsd
  3. Jan 6, 2009 #2


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    What do you mean, you don't know where it ends?
    Isn't the (r, theta, phi) system relative to (x, y, z)?
  4. Jan 6, 2009 #3
    May be I am missing something very simple here. But I do not know the [tex](r,~\theta,~\phi)[/tex] as well. I did try to do it that way though to start with.

    All I know is the magnitudes in the [tex]\hat{r},~\hat{\theta}[/tex] and [tex]\hat{\phi}[/tex] directions and the starting point. And nothing else.

    Can the [tex](r,~\theta,~\phi)[/tex] be found out from the magnitudes in the [tex]\hat{r},~\hat{\theta}[/tex] and [tex]\hat{\phi}[/tex] ([tex]E_r,~E_{\theta},~E_{\phi}[/tex] as above)?

    It can be really simple. But I cannot just think about it right.
  5. Jan 6, 2009 #4


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    Yes, you might be missing something or I might.
    But aren't the coordinate values simply the coefficients of the unit vectors?
    Like, in a Cartesian system you can write either (3, 0, -2) for the coordinates of a point, or you can describe it by a vector [itex]3 \hat x + 0 \hat y - 2 \hat z[/itex].
    When writing down a tuplet of numbers like (3, 0, -2), we are implicitly assuming that we have these three basis vectors [itex]\hat x, \hat y, \hat z[/itex] and we are using them to fix our point.
  6. Jan 6, 2009 #5
    Thanks Compuchip. I got your point. So stupid of me. The (x,y,z) can be used to find theta, phi. My bad.
  7. Jan 7, 2009 #6


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    You're welcome. We all get confused sometimes.
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