How to use geometry to express unit vectors of spherical coordinate system in terms of Cartesian unit vectors

Question

It's quite easy to express unit vector $\hat{r}$ in sum linear combinations of Cartesian unit vectors $\hat{x}$, $\hat{y}$ and $\hat{z}$. But I am not sure how I can use geomtery to find a Cartesian sum for $\hat{\theta}$ and $\hat{\phi}$. Can anyone show how this is possible?

Answer 1

Here's an easy geometric way to derive the components of $\hat{\theta}$ and $\hat{\phi}$. First find $\hat{\phi}$ using cylindrical coordinates:

$$\hat{\phi}=\hat{z}\times\hat{\rho}\\ =\hat{z}\times\left(\hat{x}\cos{\phi}+\hat{y}\sin{\phi}\right)\\ =-\hat{x}\sin{\phi}+\hat{y}\cos{\phi}.$$

The expression for $\hat{\phi}$ in spherical coordinates is exactly the same as in cylindrical coordinates. Now that you know both $\hat{r}$ and $\hat{\phi}$, you can find $\hat{\theta}$ in a similar way:

$$\hat{\theta}=\hat{\phi}\times\hat{r}.$$