Trying to derive equation for acceleration in spherical coordinate system

[vweltin]

Homework Statement

I was trying to figure out how to derive acceleration in spherical coordinates, and I realized that I need to find the projection of each spherical unit vector [ e(r), e(θ), and e(φ)] onto each Cartesian unit vector [î, j, and k], but I'm not quite sure as to how to do that. It's pretty intuitive for me to calculate the projections of e(r), but the other components confuse me a bit.

Homework Equations

see attached image

The Attempt at a Solution

I've attached the work I've done so far as an image (I didn't mean for that ' near the î on the bottom to be there..)

 

[omoplata]

http://en.wikipedia.org/wiki/Unit_vectors#Spherical_coordinates" are the unit vectors in the spherical coordinate system from wikipedia. Your $\hat{e_r}$ is given as $\hat{r}$, $\hat{e_\theta}$ is given as $\hat{\theta}$ and $\hat{e_\phi}$ is given as $\hat{\psi}$.

It looks to me that your $\hat{e_\theta}$ is drawn incorrectly in the diagram. Is it drawn to go straight into the page?