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What is the sum of two vectors in spherical coordinates?

The coordinate system:

coord system

Assume we have vectors $(r_1,\theta_1,\phi_1)$ and $(r_2,\theta_2,\phi_2)$ in spherical coordinates.

I know the sum vector is not $(r_1+r_2,\theta_1+\theta_2,\phi_1+\phi_2)$ because $\hat r$, $\hat \theta$ and $\hat \phi$ are not fixed like Cartesian coordinates.

but I don't know what the sum vector will be then.

Also I know it is possible to convert to Cartesian and easily sum the components up and get the sum vector. but I want it in spherical system.

In the linked questions, the sum vector is not given in spherical coords. which is what I want.

AHB
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1 Answers1

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Converting them to Cartesian coordinates makes it easy:

$$\vec{r_1} + \vec{r_2} = \\ (r_1 \cos {\theta}_1 \sin {\phi}_1 + r_2 \cos {\theta}_2 \sin {\phi}_2) \hat{x} + \\ (r_1 \sin {\theta}_1 \sin {\phi}_1 + r_2 \sin {\theta}_2 \sin {\phi}_2) \hat{y} + \\ (r_1 \cos {\phi}_1 + r_2 \cos {\phi}_2) \hat{z} \\ =X\hat{x} + Y\hat{y} + Z\hat{z}.$$

Then you can convert back to spherical basis $(\hat{r}, \hat{\theta}, \hat{\phi})$ if you like:

$$\vec{r_1} + \vec{r_2} = \\ \sqrt{X^2 + Y^2 + Z^2}\hat{r} + \tan^{-1}\left(\frac{Y}{X}\right)\hat{\theta} + \cos^{-1}\left(\frac{Z}{\sqrt{X^2 + Y^2 + Z^2}}\right)\hat{\phi}.$$

John
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  • I know that in Cartesian coordinates. I want in spherical coords. My answer to the last part ("if you like") is I like. that''s exactly what I want! :) – AHB Dec 24 '15 at 17:01
  • Added the back conversion. – John Dec 24 '15 at 17:12