Why $\hat r \times \hat z = sin(\theta) \hat\phi$

Question

why $\hat r \times \hat z = sin(\theta) \hat\phi$ in spherical coordintates? how did the $sin(\theta)$ factor appeared?

Answer 1

[I think that this depends on the exact way you choose your spherical polars.

We have that $$\hat{\phi}=(-\sin\phi, \cos\phi,0)$$ in Euclidean coordinates.

We also $$\hat{r}=(\sin\theta \cos\phi, \sin\theta \sin\phi, \cos\theta),\ \ \ \hat{z}=(0,0,1).$$

Calculate in usual way $$ \hat{r}\times\hat{z}=(-\sin\theta \sin\phi,\sin\theta \cos\phi,0)=\sin\theta\hat{\phi}.$$