![]() ![]() Again, should not be confused with the corresponding rotation matrix. Where is a 3 3 array representing the matrix of the components of the rotation tensor such that. When implemented as a matrix-vector operation, However, we can take advantage of the tensor representation However, the downside of this approach is that we can only write the components of a vector in just one basis, say, :Ĭonsequently, if we wish to express the rotated vector as a column vector in MATLAB, we must write it in components relative to the basis. One of the most convenient manners to represent vectors in MATLAB is to express them as column vectors. The components of the rotation tensor expressed as an array are readily compared to the components of the corresponding rotation matrix in ( 2):Īs expected, the relationship between the components of and agrees with ( 8). Expressed as a tensor, this rotation has the representation To illustrate ( 8), let us return to the example of a rotation about an axis through an angle. Because the rotation tensor is generally not assumed to be symmetric, this confusion will cause the direction of the rotation to be reversed. This matrix is easily confused with the rotation matrix. Often, the components of the rotation tensor are written in the matrix form. ![]() Comparing the final representation to the first in ( 9) yields the conclusion ( 8). Note the relabeling of the indices in the final representation. Observing that there are two equivalent representations for the transformation ,Īn alternative method to determine how the tensor components relate to the matrix components is to start with ( 5) and ( 6) and then invoke ( 4) as follows: How are the components of the tensor related to the components of the matrix ? To compare and contrast ( 6) and ( 4), one can consider a vector acted upon by. The components of the rotation tensor are defined by the representation To this end, we consider the same transformation from to expressed using a rotation tensor : It is of interest here to compare the representation ( 4) to the corresponding representation using a rotation tensor. This inverse rotation is also observed when the new basis is transformed to the original basis :Įxtrapolating from the above example, for a transformation corresponding to a rotation through an angle about an axis, we can express the basis vectors as a linear combination of the vectors forming the basis : ![]() The corresponding rotation matrix is the transpose of, denoted by. Notice that if is negative, then the direction of the rotation becomes clockwise. The transformation between the two bases is achieved by a rotation matrix and can be expressed in the following manners: Rotation tensors and their equivalent rotation matricesĬonsider a counterclockwise rotation through an angle about an axis that transforms the set of right-handed basis vectors into a new set of basis vectors. ![]()
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