Coordinate transformations Spherical to Cartesian

1. Jan 6, 2009

gaganaut

Hi,
I would like to transform a vector from Spherical to cartesian coordinate system. But the question is probably not that straight forward. :(

I have a vector say $$E = E_r~\hat{r}+E_{\theta}~\hat{\theta}+E_{\phi}~\hat{\phi}$$.

But I know only the cartesian coordinate from where it starts, say $$(x,y,z)$$ and I do not know where it ends. So I am unable to find angles $$\theta$$ and $$\phi$$ for computing the transformation matrix $$R$$ that transforms the vector $$E$$ to cartesian system. This $$R$$ is the usual matrix with sines and cosines of $$\theta$$ and $$\phi$$ and can be seen here.
http://en.wikipedia.org/wiki/Vector_fields_in_cylindrical_and_spherical_coordinates

So how do I go about it. Is there even a way to do this. Once again this is not a homework question and is for a small project that I am doing. There aren't any homeworks at this time of the year. :)

Appreciate any form of help.

Kedar

2. Jan 6, 2009

CompuChip

What do you mean, you don't know where it ends?
Isn't the (r, theta, phi) system relative to (x, y, z)?

3. Jan 6, 2009

gaganaut

May be I am missing something very simple here. But I do not know the $$(r,~\theta,~\phi)$$ as well. I did try to do it that way though to start with.

All I know is the magnitudes in the $$\hat{r},~\hat{\theta}$$ and $$\hat{\phi}$$ directions and the starting point. And nothing else.

Can the $$(r,~\theta,~\phi)$$ be found out from the magnitudes in the $$\hat{r},~\hat{\theta}$$ and $$\hat{\phi}$$ ($$E_r,~E_{\theta},~E_{\phi}$$ as above)?

It can be really simple. But I cannot just think about it right.

4. Jan 6, 2009

CompuChip

Yes, you might be missing something or I might.
But aren't the coordinate values simply the coefficients of the unit vectors?
Like, in a Cartesian system you can write either (3, 0, -2) for the coordinates of a point, or you can describe it by a vector $3 \hat x + 0 \hat y - 2 \hat z$.
When writing down a tuplet of numbers like (3, 0, -2), we are implicitly assuming that we have these three basis vectors $\hat x, \hat y, \hat z$ and we are using them to fix our point.

5. Jan 6, 2009

gaganaut

Thanks Compuchip. I got your point. So stupid of me. The (x,y,z) can be used to find theta, phi. My bad.

6. Jan 7, 2009

CompuChip

You're welcome. We all get confused sometimes.