Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

alt="images"/> can be expressed as a linear combination of , , and as shown below.

      (1.37)

Therefore, being a column matrix, can also be expressed in terms of the basic column matrices , , and by writing the following matrix version of Eq. (1.29) in .

(1.38)

      For the sake of comparing Eqs. (1.29) and (1.38) from the viewpoint of the notational logic, Eq. (1.29) is written again below.

(1.39)

      Here, it is instructive to pay attention to the interchanged location of the superscript (a) in Eqs. (1.38) and (1.39). In Eq. (1.39), must not bear (a) because it is a vector that is specified without necessarily knowing anything about the observation frame , whereas must necessarily bear (a) because it is one of the basis vectors of . In Eq. (1.38), on the other hand, must necessarily bear (a) because it represents the appearance of as observed in , whereas must not bear (a) because it is not tied up to any reference frame as explained above and expressed by Eq. (1.36).

1.6 Matrix Operations Corresponding to Vector Operations

      1.6.1 Dot Product

      Consider two vectors and , which are resolved as follows in a reference frame :

(1.40)

(1.41)

      The dot product of and can be expressed as

(1.42)

      On the other hand, according to Eq. (1.24),

      (1.43)

      Hence, Eq. (1.42) becomes

(1.44)

Owing to the definition of δ_ij, Eq. (1.44) becomes simplified to

(1.45)

      Equation (1.45) can also be written as follows in terms of and , which are the column matrix representations of and in :

(1.46)

      Equation (1.46) shows that the dot product of two vectors is equivalent to the inner product of their column matrix representations in a reference frame such as .

      1.6.2 Cross Product and Skew Symmetric Cross Product Matrices

      Consider the same two vectors and , which are expressed by Eqs. (1.40) and (1.41) as resolved in the reference frame . Their cross product can be expressed as

(1.47)