Vectors with different coordinate systems

[Hoeni]

Hi,
I am trying to simulate a freely jointed chain polymer to do that I want to put several rods (length a) on top of each other but with different angles. My problem is like this
I have a vector(1) and at the end of this vector(1) I put another vector(2), the z-axis of this vector(2)'s coordinate system is the direction of vector(1), but how do express the coordinates of the tip of this vector(2) in the coordinate system of vector(1)
t=theta f= phi a= radius
V1:
x = a sin t cos f
y = a sin t sin f
z = a cos t

and in another coordinate system
V2:
x2 = a sin t2 cos f2
y2 = a sin t2 sin f2
z2 = a cos t2

The direction of V1 is the z-axis in the coordinate system of V2
Given the fact that all the angles are known, how do I express the location of V2 in the coordinate system of V1?
Probably something with Euler transformations but i haven''t been able to figure it out.
Thanks in advance for any help.
Hoeni

 

[HallsofIvy]

You say "another coordinate system" but how are the two coordinate system related? You will need to know that before you can say anything about how descriptions of vectors in the two coordinate systems are related.

 

[Hoeni]

As I said: "The direction of V1 is the z-axis in the coordinate system of V2"
This was kind of the whole point of the question considering what I said about the freely jointed chain.

But I think I might have solved it, I used x2,y2,z2 ( derived from t2, a, and f2) and with some geometry I projected all these components on the three axes in the coordinate system of v1. My result:
x = x2 cos f + y2 cos t cos f + z2 sin t cos f
y = x2 sin f + y2 cos t sin f + z2 sin t cos f
z = 0 + y2 sin t + z2 cos t

But I need to check if I did al the calculations right.