Читать книгу Kinematics of General Spatial Mechanical Systems - M. Kemal Ozgoren - Страница 22

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 .