Deriving spherical unit vectors in terms of cartesian unit vectors

[chipotleaway]

I'm trying to find the azimuthal angle unit vector \vec{\phi} in the cartesian basis by taking the cross product of the radial and \vec{z} unit vectors.
\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)>

But the sin(\theta) shouldn't be there so we would have to multiply the cross product by 1/sin(\theta) 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?

Thanks

 

[ShayanJ]

You don't actually divide by \sin\theta. Its just that the azimuthal unit vector relies completely on the xy plane and so you should set \theta=\frac \pi 2.

 

[chipotleaway]

Thanks, that makes sense.I was following this .pdf
https://www.csupomona.edu/~ajm/materials/delsph.pdf