- #1
chipotleaway
- 174
- 0
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?
Thanks
[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?
Thanks