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(1.93)
(1.94)
(1.95)
Note that the solution obtained above is the same as the solution provided by Cramer's rule.
2 Rotation of Vectors and Rotation Matrices
Synopsis
This chapter is devoted to the rotation of vectors and the rotation operators that rotate vectors. The rotation of a vector is expressed both as a vector equation and as a matrix equation written in a selected reference frame. The vector equation is obtained as the Rodrigues formula. The matrix equation is written in terms of the rotation matrix, which is the matrix representation of the rotation operator in the selected reference frame. The expression of the rotation matrix is obtained in terms of the angle of rotation and the unit vector along the axis of rotation. It is shown that the rotation matrix can be expressed very compactly in the exponential form. This chapter also presents the salient mathematical properties of the rotation matrices that can be used conveniently in the symbolic manipulations concerning rotational kinematics. Demonstrative examples are also included.
2.1 Vector Equation of Rotation and the Rodrigues Formula
Figure 2.1 illustrates the rotation of a vector
Incidentally, a vector may be acted upon, simultaneously or successively, by two kinds of displacement operators. One of them is a rotation operator, which is defined as an operator that changes only the orientation of a vector irrespective of any possible change in its location. The other one is a translation operator, which is defined as an operator that changes only the location of a vector without changing its orientation.
As mentioned above, a rotation operator is not affected by any translational displacement. Therefore, without any loss of generality, the rotation of
Figure 2.1 Rotation of a vector about an axis.
The rotation illustrated in Figure 2.1 can be described briefly in one of the following ways, in which the rotation operator is denoted by
(2.1)
(2.2)
The resultant vector
Referring to Figure 2.1, the vectors
In Eqs. (2.3) and (2.4),
(2.5)
Referring again to Figure 2.1, the vector
