Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

      (2.95)

      The corresponding vector equations can be written as follows, like those written for and :

      (2.96)

      (2.97)

      2.8.2 Example 2.2

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)

      2.8.3 Example 2.3

Equation (2.76) is verified here as a follow‐up to Example 2.2.

      Starting with Eq. (2.75), the powers of can be expressed as follows:

      Thus, for all k ≥ 0, it happens that

(2.106)

      On the other hand, the following Taylor series expansion can be written for .

(2.107)

Using Eq. (2.106) and noting that , Eq. (2.107) can be written in the following factorized form.

(2.108)

Hence, as the verification of Eq. (2.76), Eq. (2.108) implies that

      (2.109)

      2.8.4 Example 2.4

In this example, Eq. (2.87) will be verified for ijk = 123 as explained below. However, it can be verified similarly for any other distinct (unrepeated) triplet of indices as well.

      For ijk = 123, Eq. (2.87) becomes

(2.110)

Let be the product of the three matrices on the left‐hand side of Eq.