Читать книгу Kinematics of General Spatial Mechanical Systems - M. Kemal Ozgoren - Страница 39

Equation (2.75) is verified here as an example.

Consider three vectors represented by the column matrices , , and in a reference frame . Let these vectors be such that

(2.98)

Let the same vectors be rotated by the rotation operator represented by so that

(2.99)

(2.100)

(2.101)

Since a rotation operator retains the cross product relationship, the new vectors also satisfy an equation similar to Eq. (2.98). That is,

(2.102)

Using Eqs. (2.99)–(2.101), Eq. (2.102) can be manipulated as follows:

(2.103)

Hence, it is seen that

(2.104)

Equations (2.99) and (2.104) imply the following mutual correspondence, which verifies Eq. (2.75) when and are replaced with and , respectively.

(2.105)