rotation operator associated with
is represented in
by the matrix
, which is designated as the
kth
basic rotation matrix. It is expressed as follows:
(2.28)
Referring to Section for the discussion about the basic column matrix , it is to be noted that, just like , the basic rotation matrix is also an entity that is not associated with any reference frame. This is because represents the rotation operator in its own frame , whatever is. In other words,
(2.29)
By using Eqs., can be expressed in three equivalent ways as shown in the following equations.
(2.30)
(2.31)
(2.32)
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 is first rotated into a vector and then is rotated into another vector . These two successive rotations can be described as indicated below.
(2.36)
On the other hand, according to Euler's theorem, the rotation of into can also be achieved directly in one step. That is,
(2.37)
The following matrix equations can be written for the rotational steps described above as observed in a reference frame .
(2.38)
(2.39)
(2.40)
Equations (2.39) and (2.40) show that the overall rotation matrix is obtained as the following multiplicative combination of the intermediate rotation matrices and .
(2.41)
As a general notational feature, the rotation matrix between and can be denoted by two alternative but equivalent symbols, which are shown below.
(2.42)
Although and are mathematically equivalent, their verbal descriptions are not the same. is called a rotation matrix that describes the rotation of into , whereas is called an orientation matrix that describes the relative orientation of with respect to .
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)
