# composition of rotation matrices

We wish to construct a general rotation $\mathbf{R}$ of a coordinate system by composing three elementary rotations $\mathbf{R}_1, \mathbf{R}_2, \mathbf{R}_3$, so that a vector $\mathbf{v}$ is rotated to $\mathbf{v}'$ via $$\mathbf{v}' = \mathbf{R} \mathbf{v}.$$ A common convention is to take the rotations about the $z,x,z$ axes in that order, such that the rotations are intrinsic; i.e., the axes of rotation are (in order) $z, x', z''$, where the primes indicate that the coordinate axis has been rotated: $$\{x,y,z\} \overset{\mathbf{R}_1}{\to} \{x',y',z'\} \overset{\mathbf{R}_2}{\to} \{x'',y'',z''\} \overset{\mathbf{R}_3}{\to} \{x''',y''',z'''\}.$$ In this case, $$\mathbf{R} = \mathbf{R}_3 \mathbf{R}_2 \mathbf{R}_1$$ and the elementary rotations are given by $$\mathbf{R}_1= \begin{bmatrix} \cos \phi & \sin \phi & 0 \\ -\sin \phi & \cos \phi & 0 \\ 0 & 0 & 1 \end{bmatrix} \quad \mathbf{R}_2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \theta & \sin \theta \\ 0 & -\sin \theta & \cos \theta \end{bmatrix} \quad \mathbf{R}_3= \begin{bmatrix} \cos \psi & \sin \psi & 0 \\ -\sin \psi & \cos \psi & 0 \\ 0 & 0 & 1 \end{bmatrix}.$$

The aim is to first rotate in the $xy$-plane by $\phi$, achieved via $\mathbf{R}_1$, then to rotate in the $z'y'$-plane by $\theta$, achieved via $\mathbf{R}_2$, and finally rotate in the $x''y''$-plane by $\psi$, achieved via $\mathbf{R}_3$.

Question: Why does this sequence of rotations effect an intrinsic rotation, instead of an extrinsic one? $\mathbf{R}_2$, for example, looks like it effects a rotation in the $yz$-plane, not the $y'z'$-plane. However, this example appears in two textbooks (Thornton and Marion 2004, Goldstein 1980) as an intrinsic rotation. Both texts are very clear that the sequence of rotations is intrinsic. All matrices are given explicitly as I have reproduced above.

• It depends on whether the rotation is applied on the left to column vectors or on the right to row vectors. – Rahul May 31 '14 at 22:16
• What is the order of the angles? Because of your last comment, it seems to me that they've converted between intrinsic and extrinsic rotations. You could hardly tell the difference unless you know the order in which the rotation angles should go. – Muphrid May 31 '14 at 22:22
• @Rahul I have modified the question to reflect that the rotation is to be applied to the left on column vectors. – Eric Kightley Jun 1 '14 at 4:14
• @Muphrid I have modified the question to indicate the order of the angles and emphasize that there is no intended conversion between extrinsic and intrinsic rotations. – Eric Kightley Jun 1 '14 at 4:15
• Then as far as I can see, the order of the matrices is wrong. The only intrinsic rotation this product of matrices produces is one in which the $\psi$ rotation is performed first. – Muphrid Jun 1 '14 at 4:18