# Showing relationship between cartesian and spherical unit vectors

#### skatenerd

##### Active member
I am asked to show that when $$\hat{e_r}$$, $$\hat{e_\theta}$$, and $$\hat{e_\phi}$$ are unit vectors in spherical coordinates, that the cartesian unit vectors
$$\hat{i} = \sin{\phi}\cos{\theta}\hat{e_r} + \cos{\phi}\cos{\theta}\hat{e_\phi} - \sin{\theta}\hat{e_\theta}$$
$$\hat{j} = \sin{\phi}\sin{\theta}\hat{e_r} + \cos{\phi}\sin{\theta}\hat{e_\phi} - \cos{\theta}\hat{\theta}$$
$$\hat{k} = \cos{\phi}\hat{e_r} - \sin{\phi}\hat{e_\phi}$$
I'm having a bit of trouble figuring out where to start with this. I totally understand geometrically how to convert $$x$$ $$y$$ and $$z$$ coordinates to spherical, but I feel like that isn't helping me here. How do I begin to find the spherical unit vectors in terms of the cartesian unit vectors, or vise versa?

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Re: showing relationship between cartesian and spherical unit vectors

I am asked to show that when $$\hat{e_r}$$, $$\hat{e_\theta}$$, and $$\hat{e_\phi}$$ are unit vectors in spherical coordinates, that the cartesian unit vectors
$$\hat{i} = \sin{\phi}\cos{\theta}\hat{e_r} + \cos{\phi}\cos{\theta}\hat{e_\phi} - \sin{\theta}\hat{e_\theta}$$
$$\hat{j} = \sin{\phi}\sin{\theta}\hat{e_r} + \cos{\phi}\sin{\theta}\hat{e_\phi} - \cos{\theta}\hat{\theta}$$
$$\hat{k} = \cos{\phi}\hat{e_r} - \sin{\phi}\hat{e_\phi}$$
I'm having a bit of trouble figuring out where to start with this. I totally understand geometrically how to convert $$x$$ $$y$$ and $$z$$ coordinates to spherical, but I feel like that isn't helping me here. How do I begin to find the spherical unit vectors in terms of the cartesian unit vectors, or vise versa?
Well, let's see...

If we take a unit vector in the direction of r, we get:
$$\hat{e_r} = \frac{x\hat{i} + y\hat{j} + z\hat{k}}{\sqrt{x^2+y^2+z^2}}$$
Yes?

Perhaps we can create a set of 3 equations like this?
Can you find an equation for $\hat{e_\phi}$ and $\hat{e_\theta}$?

After that we can try and solve it for the cartesian unit vectors!