Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

      Since and are orthogonal, . Therefore, Eq. (1.77) gives γ as

      (1.78)

      Hence, and are obtained as shown below.

      (1.79)

      (1.80)

      1.8.3 Example 1.3

      Consider the following 3 × 3 matrix equation, which is to be solved for .

(1.81)

      The matrix can be expressed as follows in terms of its columns.

      (1.82)

      Along with , can be expressed as follows in terms of its elements.

      (1.83)

      Hence, Eq. (1.81) can be written in a more detailed form as

(1.84)

      Equation (1.84) leads to the following scalar equations with the indicated premultiplications.

(1.85)

      (1.86)

(1.87)

      Note that, for i ∈ {1, 2, 3} and j ∈ {1, 2, 3},

      (1.88)

      Thus, Eqs. (1.85)–(1.87) reduce to the following equations.

(1.89)

      (1.90)

(1.91)

      Equations (1.89)–(1.91) imply that

      (1.92)

      Therefore, if , Eqs. (1.89)–(1.91) give the elements of

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