Velocity vectors in three dimensions

[Mattew]

Hi everybody,
I have a quite simple question (in my opinion) but my background is quite poor about three dimensions physics.
I need to express two velocity vectors, v1 and v2, in three dimensions polar coordinates, which means using polar and azimuthal angles. The two polar angles represent enough information to define the velocity vector?
Then what I'm concerned about is how to do operations between theese two vectors: in particular I'd have to compute the relative velocity of the two objects moving according to the velocity vectors above, how can I do it using sin and cos of the polar angles (teta1,phi1) of v1 and (teta2,phi2) of v2?
Thanks in adavance for your time and help

 

[Defennder]

Have you tried expressing the velocity vectors, presumably already given in rectangular cartesian unit vectors (i, j, k) in mutually orthogonal unit vectors in the spherical coordinate system? Or do you have trouble doing that?

 

[Mattew]

Let me see if I get it: in this coordinate system the location of a point p in the sphere is defined by a triple (r,teta,phi), where r is the distance of the point from the origin, teta is the angle between the vector connecting p to the origin and the z-axis, and phi is the angle between the projection of p on the x-y plane and the x-axis. Is that right?
Having that, in this system a velocity vector could be defined by r as its magnitude, and the teta/phi as the direction on movement, isn't it?
Now, if we have v1=(r1,teta1,phi1) and v2=(r2,teta2,phi2), how do I compute the relative velocity of the two objects?
I hope I'm not too confusing...

 

[Dick]

You compute v1-v2. This is hard to do directly in spherical coordinates - so convert to (x,y,z) first. Then you can just subtract component by component - then convert back to spherical coordinates if you need to.

Eg. z=r*cos(theta)
x=r*sin(theta)*cos(phi)
y=r*sin(theta)*sin(phi)

 

[Mattew]

Thanks, I was looking for a way to do it without introduncing x,y,z at all. I actually need to express it in a proof by means of the spherical angles and magnitude, but I need no computations...only the relative velocity as a function of the original spherical angles of v1 and v2 (let's immagine we got the two vectors (and relative angles, teta/phi) drawn in a 3d coordinate system and we want to paint the relative velocity velocity vector v1-v2 with the angles directly by sin/cos of teta1/phi1 and teta2/phi2-no change of coordinate system.