Can Vector Projection Solve Angular Displacement in Orbit Calculations?

[trmcclain]

Homework Statement

I don't know if its possible to do this, but I ran into this problem when trying to right a program which creates orbits for individual points. Anyways...
Given 2 position vectors (tails anchored to the origin), say A and B and the angle between them $$\theta$$, A is unknown.
I'm going off the assumption that |A| can be less than |B|. We know the vector B and the angle between them $$\theta$$. And I need to recover A[\B].

Homework Equations

I thought maybe I could do a projection, but that wouldn't give me back what I needed. The relevant equations I would think would be anything having to do with triangle geometry.

The Attempt at a Solution

Attaching pictures of what I did. If anything is too fuzzy, please let me know and I'll try to give computerized diagrams.

 

[micromass]

You'll need to rotate B by a angle $$\theta$$ it seems...

A rotation by $$\theta$$ can be given by the matrix

$$\left(\begin{array}{cc} <br /> \cos(\theta) & -\sin(\theta)\\<br /> \sin(\theta) & \cos(\theta)<br /> \end{array}\right)$$

 

[trmcclain]

But that only gives be B rotated to some angle $$\theta$$
I fail to see how that will help me figure out the magnitude of A

 

[micromass]

Do you know any other thing about A. Do you just know that A and B have angle theta? Then the magnitude of A can be everything.

Or do I need to watch the diagrams you posted. To be honest, they were to fuzzy, so I ignored them. Is there anything important on them?

 

[trmcclain]

Important? I guess not really, they just showed what I tried. Maybe some more background as to the program I'm writing might help. It's for a project in graduate school, I'm trying to see what the effects of artificially inducing charge on a cloud of massive particles are, so kind of like playing God and giving the stars of a globular cluster some amount of charge and seeing what they're orbits are like.

In trying to account for the fact that their orbits may or may not be circular, I posted the question onto this forum.
For every iteration of time in the program I'm writing, I calculate the angle each star has moved. Knowing each star's initial position and the randomized angular velocity I give it, I try to calculate the position of the next iteration for the next round of calculations.
So at the get go, I have the intial position R and the initial angle $$theta$$ and the initial angular velocity. Using the initial angular velocity and some arbitray amount of time, I calculate the new $$theta$$ for that time but how would I use that to find the new position R?

 

[micromass]

But isn't every orbit circular? I mean, if you only have the initial position R and the angle theta, then you can't figure out what the new position is. You do know the initial position up to a multiple. But you do have to know something about the magnitde of the new position...

 

[trmcclain]

every orbit isn't necessarily circular, especially in globular clusters. Mostly they'll be precessing elliptical orbits, but I was told I can't assume elliptical-ness from the get go, I have to watch it evolve.

So basically what I'm getting is that there is no trig or geometric relation that can tell me the new position given an angular displacement and a position. That means there has to be something in the math that can tell me, some equation or derivation. Nuts...