Definition:Rotation Matrix

Definition
Let $\tuple {\mathbf e_1, \mathbf e_2, \mathbf e_3}$ be the orthonormal basis of a Cartesian coordinate system $C$ on (ordinary) space $S$.

Let $O$ be the origin of $C$.

Let $\tuple {\mathbf e'_1, \mathbf e'_2, \mathbf e'_3}$ be the orthonormal basis of another Cartesian coordinate system $C'$ on $S$, also with origin $O$ and with the same orientation as $C$.

The rotation matrix $R$ from $C$ to $C'$ is the square matrix of order $3$:


 * $R = \begin {pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end {pmatrix}$

where $a_{i j}$ is defined as:
 * $\mathbf e'_i = \paren {\mathbf e'_i \cdot \mathbf e_j} \mathbf e_j \equiv a_{i j} \mathbf e_j$

for $i, j \in \set {1, 2, 3}$.