# 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.