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Deriving spherical unit vectors in terms of cartesian unit vectors

  1. Oct 27, 2014 #1
    I'm trying to find the azimuthal angle unit vector [itex]\vec{\phi}[/itex] in the cartesian basis by taking the cross product of the radial and [itex]\vec{z}[/itex] unit vectors.
    [itex]\vec{z} \times \vec{r} = <0, 0, 1> \times <sin(\theta)cos(\phi), sin(\theta)sin(\phi), cos(\theta)> = <-sin(\theta)sin(\phi), sin(\theta)cos(\phi), 0)>[/itex]

    But the [itex]sin(\theta)[/itex] shouldn't be there so we would have to multiply the cross product by [itex]1/sin(\theta)[/itex] to get the correct unit vector. But why do we need to do this if the magnitude is already one?

    Also, how would you do this using trigonometry?

  2. jcsd
  3. Oct 28, 2014 #2


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    Gold Member

    You don't actually divide by [itex] \sin\theta [/itex]. Its just that the azimuthal unit vector relies completely on the xy plane and so you should set [itex] \theta=\frac \pi 2 [/itex].
  4. Oct 28, 2014 #3
    Thanks, that makes sense.I was following this .pdf
    https://www.csupomona.edu/~ajm/materials/delsph.pdf [Broken]
    Last edited by a moderator: May 7, 2017
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