# Thread: Determining a rotation in R^3 given angles of new axes w.r.t. original x-axis

1. ## Determining a rotation in R^3 given angles of new axes w.r.t. original x-axis

The problem is essentially as follows:

In $\mathbb{R}^3$, a rotated coordinate system's axes are at angles $\phi_1$, $\phi_2$ and $\phi_3$ to the x-axis in standard Cartesian coordinates. Find the rotation matrix between the two coordinate systems.

I've tried to attack this both algebraically, using the 3 given direction cosines and orthonormality properties of the rotation matrix, as well as geometrically, trying to use trigonometric relationships between the two coordinate systems, but just can't seem to make it work. Algebraically it gets really messy really quickly and I can't solve for the unknowns and geometrically I can't work out enough useful relationships.

Any help would be much appreciated. Cheers!

2. ## Re: Determining a rotation in R^3 given angles of new axes w.r.t. original x-axis

Defining a rotation with angles to the original x-axis is a bit unusual. Do you have a specific reason for using these kind of angles? Much more common is the use of Euler angles.

3. ## Re: Determining a rotation in R^3 given angles of new axes w.r.t. original x-axis

I know that this is non-standard and that Euler angles, quaternions, etc. are the more usual way of paramaterizing such rotations. Though in theory, as SO(3) is 3-dimensional, any arbitrary parameterization is described by 3 numbers as in the problem.

It comes from here: http://bit.ly/qBH13I, 6.ii).