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