title: 3d rotations defined by angles between original and new coordinate axes
body: <p>This type of question has been asked before, but I haven't been able to find a satisfactory explanation. Perhaps I'm just a bit slow...</p>
  <p>Say that I have a global right-handed coordinate system represented by X, Y, and Z. Say then that I have a rotated cube whose lower, left, back corner defines the origin of a new right-handed coordinate system and whose sides connecting to the origin define the X', Y', and Z' axes. Next we define the angle between X and X' as alpha, the angle between Y and Y' as beta, and the angle between Z and Z' as gamma.</p>
  <p>I would like to be able to use rotation matrices and the angles alpha, beta, and gamma to define the coordinates of the rotated cube in the X, Y, and Z global coordinate system. These rotation matrices are given by:</p>
  <p>Rx = [[1, 0, 0], [0, cos(alpha), -sin(alpha)], [0, sin(alpha), cos(alpha)]]</p>
  <p>Ry = [[cos(beta), 0, sin(beta)], [0, 1, 0], [-sin(beta), 0, cos(beta)]]</p>
  <p>Rz = [[cos(gamma), -sin(gamma), 0], [sin(gamma), cos(gamma), 0], [0, 0, 1]]</p>
  <p>For example, a rotation about the z-axis for a vector X in R3 gives a new vector x by:</p>
  <p>x = Rz * X</p>
  <p>These rotation matrices work perfectly well and order of matrix multiplication does not matter when two of the three rotation angles are zero e.g. when alpha≠0 and beta=gamma=0 or beta≠0 and alpha=gamma=0 or gamma≠0 and alpha=beta=0. This is because when the rotation angle for a particular rotation matrix is equal to 0 the rotation matrix becomes the identity matrix.</p>
  <p>The order of these matrices matters when more than one of the rotation angles are nonzero. My understanding is that for the following rotation:</p>
  <p>x = Rz * Ry * Rx * X</p>
  <p>represents a rotation about the x-axis, then a rotation about the new y-axis, and then a rotation about the new new z-axis.</p>
  <p>Is there an order of these rotations that does what I would like? Or a relation in order to do so? In summary, I would like a relation to be able to rotate these based only on the angles between X' and X, Y' and Y, and Z' and Z.</p>