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rotation matrix can be differentiated with respect to θ as follows:

      (2.73)

      1  Rotation About Rotated Axis

      Let be rotated into by so that

(2.74)

Then, it can be shown that Eq. (2.74) leads to the following equations.

(2.75)

(2.76)

Equation (2.76) is the expression of the rotation about rotated axis formula.

      1  Shifting Formulas for the Rotation Matrices

      The following two formulas, which are called shifting formulas, can be obtained as two consequences of Eq. (2.76).

      (2.77)

      (2.78)

      2.7.2 Mathematical Properties of the Basic Rotation Matrices

In the following formulas, σ_ijk is defined as in Chapter 1. That is, for distinct indices only,

      (2.79)

      1  Expansion Formulas

      If j ≠ i,

      (2.80)

      (2.81)

      If j = i,

      (2.82)

      1  Shifting Formulas with Quarter and Half Rotations

      If j ≠ i,

(2.83)

      (2.84)

      If j = i,

      (2.85)

(2.86)

      1  Three Successive Half Rotations About Mutually Orthogonal Axes

      Provided that i ≠ j ≠ k,

(2.87)

2.8 Examples Involving Rotation Matrices

      2.8.1 Example 2.1

      The first basis vector of a reference frame is rotated successively in two different sequences, which are indicated below. It is required to express the resultant vectors in .

      (2.88)

      (2.89)

      In the first sequence, and are obtained as described below.

(2.90)

(2.91)

Noting that , the vector equations corresponding to Eqs. (2.90) and (2.91) can be written as follows:

      (2.92)

      (2.93)

      In the second sequence, and are obtained as described below.

      (2.94)

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