= Composition of rotation matrices
{title2=$v''=R_2R_1v$}
Apply $R_1$ to a column <coordinate vector>, then apply $R_2$ in the same fixed <basis>. The <function composition> is
$$
v''=R_2(R_1v)=(R_2R_1)v.
$$
Thus the rightmost <matrix> acts first. The product is again a <rotation matrix>, since $(R_2R_1)^T(R_2R_1)=I$ and $\det(R_2R_1)=1$. For <planar rotations> about the same origin, the <sine addition formula> and <cosine addition formula> give
$$
R_\beta R_\alpha=R_{\alpha+\beta}=R_\alpha R_\beta.
$$
For a concrete <composition of rotation matrices>, first rotate $(2,0)^T$ by $\pi/3$ <radians>, reaching $(1,\sqrt3)^T$, then by $\pi/6$ <radians>:
$$
\begin{pmatrix}\sqrt3/2&-1/2\\1/2&\sqrt3/2\end{pmatrix}
\begin{pmatrix}1\\\sqrt3\end{pmatrix}
=\begin{pmatrix}0\\2\end{pmatrix}.
$$
The total <angle> is $\pi/2$. Reversing these two <planar rotations> gives the same answer, but <noncommuting rotations about fixed axes> show why one cannot assume that property for <rotations in three dimensions>.<Modern Robotics: Mechanics, Planning, and Control by Kevin M. Lynch and Frank C. Park>[Modern Robotics, §3.2.1.1, propositions 3.4–3.5, preprint p. 69: closure and the distinction between planar and spatial composition]{ref}
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