(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I'm doing a problem that involves expressing, for two arbitrary vectors [itex]\vec{x}[/itex] and [itex]\vec{x'}[/itex],

[tex]|\vec{x}-\vec{x'}|[/tex]

in spherical coordinates ([itex]\rho,\theta,\phi[/itex]).

2. Relevant equations

Law of Cosines:

[tex]c^{2}=a^{2}+b^{2}-2ab\cos\gamma[/tex]

where [itex]\gamma[/itex] is the angle between a and b.

3. The attempt at a solution

Using the law of cosines, we can write

[tex]|\vec{x}-\vec{x'}|=(\rho^{2}+\rho'^{2}-2\rho\rho'\cos\gamma)^{\frac{1}{2}}[/tex]

but I can't figure out what the angle between the vectors [itex]\gamma[/itex] would be. I imagine a plane formed by the two vectors, and the angle being between the two vectors in that plane. How I describe this mathematically, though, is confusing. Can anyone push me in the right direction?

EDIT:

Alright, I took a new approach. I decided to express the difference in Cartesian coordinates, and then convert to spherical coordinates. In doing so, I get:

[tex]|\vec{x}-\vec{x'}|=(\rho^{2}+\rho'^{2}-2\rho\rho'[\sin\theta\sin\theta'(\cos\phi\cos\phi'+\sin\phi\sin\phi')+\cos\theta\cos\theta'])^{\frac{1}{2}}[/tex]

which gives me the [itex]\gamma[/itex] I was looking for. It would be nice if there were an easy way to simply that, though...

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# Homework Help: Difference between Two Vectors, Spherical Coordinates

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