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

      (1.83)equation

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

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

      (1.86)equation

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

      (1.88)equation

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

      (1.90)equation

      Equations (1.89)(1.91) imply that

      (1.92)equation

      Therefore, if images, Eqs. (1.89)(1.91) give the elements of images as follows:

      (1.93)equation

      (1.94)equation

      (1.95)equation

      Note that the solution obtained above is the same as the solution provided by Cramer's rule.

      Synopsis

      Incidentally, a vector may be acted upon, simultaneously or successively, by two kinds of displacement operators. One of them is a rotation operator, which is defined as an operator that changes only the orientation of a vector irrespective of any possible change in its location. The other one is a translation operator, which is defined as an operator that changes only the location of a vector without changing its orientation.

      As mentioned above, a rotation operator is not affected by any translational displacement. Therefore, without any loss of generality, the rotation of images is illustrated in Figure 2.1 so that the point O (i.e. the tail point of images) is assumed to be fixed and the rotation axis is assumed to pass through that point. Moreover, as illustrated on the left‐hand side of Figure 2.1, the vector images moves on the surface of a cone while it is rotated into the vector images. The projected appearance of this rotation on the base of the mentioned cone is illustrated on the right‐hand side of Figure 2.1.

Vector diagram of rotation of a vector about an axis.

      (2.1)equation

      (2.2)equation

      The resultant vector images can be expressed in terms of the initial vector images and the rotation parameters Скачать книгу