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I have two coordinate systems. The first (coordinate system 1) is at some arbitrary location and the second (coordinate system 2) is identical but for a translation along the positive z-direction of coordinate system 1 by a distance $d$. So basically, the x and y directions are aligned, just shifted upward in the z-direction by $d$.

At each origin I have a spherical coordinate system and I am trying to translate vectors in spherical coordinates from coordinate system 1 to 2. Eventually I am going to take dot products of vectors expressed in both systems.

I think it will be easiest to first translate spherical coordinates into Cartesian, and I figured that out. I do not know how to translate from one Cartesian system to the other, and vice versa. Basically, I am looking for:

$\widehat{\boldsymbol{i}}_{1} = f(\widehat{\boldsymbol{i}}_{2}, \widehat{\boldsymbol{j}}_{2}, \widehat{\boldsymbol{k}}_{2})$, $\widehat{\boldsymbol{j}}_{1} = f(\widehat{\boldsymbol{i}}_{2}, \widehat{\boldsymbol{j}}_{2}, \widehat{\boldsymbol{k}}_{2})$, $\widehat{\boldsymbol{k}}_{1} = f(\widehat{\boldsymbol{i}}_{2}, \widehat{\boldsymbol{j}}_{2}, \widehat{\boldsymbol{k}}_{2})$,

$\widehat{\boldsymbol{i}}_{2} = f(\widehat{\boldsymbol{i}}_{1}, \widehat{\boldsymbol{j}}_{1}, \widehat{\boldsymbol{k}}_{1})$, $\widehat{\boldsymbol{j}}_{2} = f(\widehat{\boldsymbol{i}}_{1}, \widehat{\boldsymbol{j}}_{1}, \widehat{\boldsymbol{k}}_{1})$, $\widehat{\boldsymbol{k}}_{2} = f(\widehat{\boldsymbol{i}}_{1}, \widehat{\boldsymbol{j}}_{1}, \widehat{\boldsymbol{k}}_{1})$

I know the origin of coordinate system 2 is at $0 \widehat{\boldsymbol{i}}_{1} + 0 \widehat{\boldsymbol{j}}_{1} + d \widehat{\boldsymbol{k}}_{1}$, but when I try to write out formulas, I get contradictory results.

Thank you to anyone who can help. I should have paid better attention in Calc III.

Brad
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  • I am a bit unclear on what you are asking. Vectors have only two attributes: magnitude and direction. Neither attribute is affected by a translation. – John Wayland Bales Oct 13 '18 at 22:55
  • If you have $\mathbf{i}_1=(1,\theta,z)$ in polar coordinates then $\mathbf{\hat{i}}_1=(1,\theta,z-d)$. – John Wayland Bales Oct 13 '18 at 23:05
  • See here: https://math.stackexchange.com/questions/243142/what-is-the-general-formula-for-calculating-dot-and-cross-products-in-spherical – John Wayland Bales Oct 13 '18 at 23:37
  • I have two points, each with their own coordinate systems, in my problem. I should be able to write the components of any vector in the basis of either system. I’m trying to figure out how to convert from one to the other. Unfortunately the polar coordinates aren’t helpful to me. And I know already how to go from spherical coordinates to Cartesian for a common origin. I don’t know how to convert to the second origin. – Brad Oct 14 '18 at 00:40

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