Subtracting vectors from different origins


Joined Mar 6, 2009
I'm pretty sure that interpreting the spherical vector coefficients as a magnitude and two angles is not consistent with the adopted convention.

Suppose you had a vector (the one I'm working on at the moment)

\(\bf A=0.535 \hat{\bf x}-2.54 \hat{\bf z}\)

How would one express the equivalent vector in spherical polar coordinates at a point P(1,3,5)?

One approach would be to form a conversion matrix

\( \left[ \bf M \right ] =\left [ \begin{array}{cc} sin(\theta)cos(\phi) & sin(\theta)sin(\phi) & cos(\phi) \\ cos(\theta)cos(\phi) & cos(\theta)sin(\phi) & -sin(\theta) \\ -sin(\phi) & cos(\phi) & 0 \end{array}\right ] \)


\(\theta \ and \ \phi\)

are part the spherical coordinates of P.


One would substitute

\(\theta=32.3^o=0.564 rad\) and \( \phi=71.6^o=1.25 rad\)

to evaluate the terms for matrix M

To obtain

\( \left[ \bf M \right ] =\left [ \begin{array}{cc} 0.169 & 0.507 & 0.845 \\ 0.267 & 0.802 & -0.535 \\ -0.949 & 0.316 & 0 \end{array}\right ] \)

The spherical polar form coefficients of A at P would then be

\(\bf M \left [ 0.535 \\ 0 \\ -2.54 \right ]=\left [ -2.06 \\ 1.5 \\ -0.507\right ]\)

Giving A in spherical form as

\(\bf A=-2.06\hat{\bf \rho}+1.5\hat{\bf \theta}-0.507\hat{\bf \phi}\)

As far as I know this is a standard approach. Arguably the most commonly adopted one.

In reversing the process one wouldn't therefore interpret the 2nd and third coefficients of the spherical polar form of vector A as angles. All coefficients are rectilinear displacements.

An important cross-check would be to evaluate that the vector magnitude value is consistent between the two forms - Cartesian & Spherical

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Joined Jul 7, 2009
t_n_k, I must recant my interpretation as flawed and agree that your approach is likely the correct one. I'm so used to thinking in spherical coordinates with angles; but it appears this problem really is dealing with the vector's components as linear dimensions.

OP, ignore my posts and concentrate on t_n_k's approach. I'm gonna have to revisit this when I get time -- it looks like a gap in my knowledge (one among many as my wife would say :p). Thanks for pointing this out to me t_n_k. Can you recommend a good online reference?


Joined Mar 6, 2009

Until this thread started I was of an identical opinion to yours. So we were both under a misapprehension.

I admire the way you acknowledge your wife and her influence in your life. She must be a woman of true worth.

Try the first link below which actually has some worked examples. If you have a means of matrix manipulation (I recollect you might use Python (??)) this can lighten the computational burden & the likelihood of making errors.

The second link is to an excellent downloadable pdf summary.

It's surprising how little comprehensive treatment of the topic there is on the web. Perhaps the mathematicians think it's so trivial it's hardly worth mentioning.

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Joined Nov 9, 2007
You might like to look at things this way.

Firstly both the vectors A and B in the problem are bound vectors, not free vectors, since they have stated points of origin or application.

There is only one spherical coordinate system mentioned and it has a different origin from the vectors A and B
This means that addition and subtraction only makes sense in certain circumstances.

For instance you could regard them as representing a displacement field, whereby the whole grid is moved along their respective lines of action a distance equal to their magnitude.

eg A could be the instruction "standing at the corner of the building walk 20 paces East".

Since we are talking subtraction we add minus B so if B is the instruction walk 20 paces North -B is the instruction walk 20 paces South.

A - B represents √(400 + 400) paces South East


Joined Feb 6, 2012
I've found the answer but I still don't understand what's the use of (1, pi/2, 0) and (3, pi/2, pi/2)? I converted it, but I didn't find the use of them. Would you tell me please? :)