Rotation as a vector
What exactly do you mean when you say you have the angular position of an object? In orthogonal 3D space, you can use two angles and a distance along an axis or one angle and two distances along axes to define a point in space. The three parts (2 angles, 1 distance or 1 angle and two distances) make up the position vector. If, when you say angular position, you mean just the angular parts of those vectors, you will not be defining a single point in space. I assume this is not the case. Remember that your angular position is at what angle you rise from a plane to point at your object. It still doesn't tell you how far that object is away from you.
You surely can project the position vector into the new space. You can find the equations for axis rotation in many math and engineering texts.