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Upon inserting the expressions of the basic column matrices into Eq. (2.30), the basic rotation matrices can be expressed element by element as shown below.
(2.33)
(2.34)
(2.35)
2.5 Successive Rotations
Suppose a vector
(2.36)
On the other hand, according to Euler's theorem, the rotation of
(2.37)
The following matrix equations can be written for the rotational steps described above as observed in a reference frame
(2.38)
Equations (2.39) and (2.40) show that the overall rotation matrix
(2.41)
As a general notational feature, the rotation matrix between
(2.42)
Although
In a case of m successive rotational steps, the following equations can be written by using the alternative notations described above.
(2.43)
(2.44)
(2.45)
2.6 Orthonormality of the Rotation Matrices
Suppose a vector
(2.46)
Since a rotation operator does not change the magnitude of the vector it rotates, the following equations can be written.
Equation (2.47) implies that
Hence, Eq. (2.48) implies further that
