MHB Showing relationship between cartesian and spherical unit vectors

[skate_nerd]

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?

 

[I like Serena]

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!