Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren
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(1.41)
The dot product of
(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 δij, Eq. (1.44) becomes simplified to
(1.45)
Equation (1.45) can also be written as follows in terms of
(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
(1.47)
On the other hand, according to Eq. (1.25),
(1.48)
Hence, Eq. (1.47) becomes
(1.49)
Equation (1.49) implies that
(1.50)
By using the definition of εijk given by Eq. (1.26), Eq. (1.49) can be worked out to what follows:
(1.51)
Upon comparing the coefficients of the basis vectors of
(1.52)
Equation (1.52) can be written again as follows by factorizing the column matrix expression on its right side:
(1.53)
Furthermore, Eq. (1.53) can be written compactly as
(1.54)
In Eq. (1.54),
Considering an arbitrary column matrix
(1.55)
The inverse of the ssm operator is the colm (column matrix) operator, which is defined so that
(1.56)
Coming back to the cross product operation,