Combined symmetrical transformations of vectors

Unrestricted transformations of a vector can, therefore, be represented using a sequence of matrix-vector transformations by first, evaluating the product of the rotation matrix and the original vector as shown in Eq. 1.32 and second, evaluating the sum of the obtained vector and the corresponding translation vector, as shown in Eq. 1.35. The combined unrestricted transformation is, therefore, represented in the expanded form using Eq. 1.38, or using the compact form as shown in Eq. 1.39. 

It is practically obvious that simultaneously or separately acting rotations (either proper or improper) and translations, which portray all finite and infinite symmetry elements, i.e. rotation, roto-inversion and screw axes, glide planes or simple translations can be described using the combined transformations of vectors as defined by Eqs. 1.38 and 1.39. When finite symmetry elements intersect with the origin of coordinates the respective translational part in Eqs. 1.38 and 1.39 is 0, 0, 0 and when the symmetry operation is a simple translation, the corresponding rotational part becomes unity, E, where 

At this point, the validity of Eqs. 1.38 and 1.39 has been established when rotations were performed around an axis intersecting the origin of coordinates. We now establish their validity in the general case by considering vector X and a symmetry operation that includes both the rotational part, R, and translational part, t. Assume that the symmetry operation is applied in a crystallographic basis where the rotation axis is shifted from the origin of coordinates by a vector At. 

we select a new basis in which the rotation axis intersects with the origin of coordinates. This is equivalent to changing the coordinates of the original vector from X to x, where 

According to Eq. 1.39, the symmetry transformation in the new basis results in vector x, where 

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