Rotation of vectors

Suppose that we rotate the Cartesian axes counterclockwise through an angle a. A two-dimensional vector v = (at, y) is transformed in this way to v = (x /) through the relationship 

Any ordinary vector (e.g. a three-component Cartesian vector) transforms in a similar way 

Any rotation in three dimensions can be specified by three angles. It is customary to use the Euler angles a, p and y, defining the rotation in three steps  

Then the effect of rotation is characterised by the rotation matrix 

A scalar quantity is invariant under rotation. Other (vector, tensor) quantities transform like products of coordinates, e.g. x2, xy, etc. 

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Figure 1 The two diagrams show rotation of vectors equal in magnitude, on the left, and unequal vectors differentiated by absorption, on the right. The two situations lead to ORD and CD respectively.

Note that the novel linear and nonlinear terms (4.56b) can exist in ferroelectrics-antiferromagnets with developed gradient of polarization P and/or antimagnetization L. The FME coupling affects the order parameter spatial distributions, as shown in Fig. 4.32b, c. In particular, pronounced maxima appear in the polarization distribution in the regions where the gradient of the antimagnetization L exists (e.g., near the film surfaces where L and L3 alter their values due to the rotation of vector L). 

By rotation of vectors oi and 02 by 45° in their plane and extension of the vector. The transformation matrix is given in the second column of Table 6.1. 

The rotation of vectors is always assumed counter clockwise. 

The molecular beam and laser teclmiques described in this section, especially in combination with theoretical treatments using accurate PESs and a quantum mechanical description of the collisional event, have revealed considerable detail about the dynamics of chemical reactions. Several aspects of reactive scattering are currently drawing special attention. The measurement of vector correlations, for example as described in section B2.3.3.5. continue to be of particular interest, especially the interplay between the product angular distribution and rotational polarization. 

Kabsch W 1978. A Discussion of the Solution for the Best Rotation to Relate Two Sets of Vectors. Acta Crystallographica A34 827-828. 

Orthogonal transfonnation of a Cartesian vector A with components Ai, T2 and. 43 in the system of ol23, under rotation of the coordinate system to nl 2 3 is expressed by the following equation... 

Field variables identified by their magnitude and two associated directions are called second-order tensors (by analogy a scalar is said to be a zero-order tensor and a vector is a first-order tensor). An important example of a second-order tensor is the physical function stress which is a surface force identified by magnitude, direction and orientation of the surface upon which it is acting. Using a mathematical approach a second-order Cartesian tensor is defined as an entity having nine components T/j, i, j = 1, 2, 3, in the Cartesian coordinate system of ol23 which on rotation of the system to ol 2 3 become... 

Orthogonal transformations preserve the lengths of vectors. If the same orthogonal transformation is applied to two vectors, the angle between them is preserved as well. Because of these restrictions, we can think of orthogonal transfomiations as rotations in a plane (although the formal definition is a little more complicated). 

The Coriolis veclor lies in the same plane as the velocity vector and is perpendicular to the rotation vector. If the rotation of the reference frame is anticlockwise, then the Coriolis acceleration is directed 90° clockwise from the velocity vector, and vice versa when the frame rotates clockwise. The Coriolis acceleration distorts the trajectory of the body as it moves rectilinearly in the rotating frame. 

The handedness (righthanded rotation of the electric vector describes clockwise rotation when looking into the beam)... 

Vectors are commonly used for description of many physical quantities such as force, displacement, velocity, etc. However, vectors alone are not sufficient to represent all physical quantities of interest. For example, stress, strain, and the stress-strain iaws cannot be represented by vectors, but can be represented with tensors. Tensors are an especially useful generalization of vectors. The key feature of tensors is that they transform, on rotation of coordinates, in special manners. Tsai [A-1] gives a complete treatment of the tensor theory useful in composite materials analysis. What follows are the essential fundamentals. 

When dealing with the motions of rigid bodies or systems of rigid bodies, it is sometimes quite difficult to directly write out the equations of motion of the point in question as was done in Examples 2-6 and 2-7. It is sometimes more practical to analyze such a problem by relative motion. That is, first find the motion with respect to a nonaccelerating reference frame of some point on the body, typically the center of mass or axis of rotation, and vectorally add to this the motion of the point in question with respect to the reference point. 

The trajectories are logarithmic spirals (Fig. 6-4). For a > 0, they wind on the singular point (i.e., the rotation of the radius vector is clockwise) for a < 0, they unwind (i.e., the rotation of the radius vector is counterclockwise). 

In quantum mechanics, angular momenta other than orbital make their appearance. Their structure is not revealed by the simple considerations leading to (7-8). That formula, in fact, arises also from the general transformation properties of vectors under rotation, as will now be shown. 

It is noteworthy that dq(e,t) does not satisfy this relation, as equality [J,x, dq] = 2 C q dq+ll (the definition of an irreducible tensor operator) does not hold for it [23]. Integration in (7.18), performed over the spherical angles of vector e, may be completed up to an integral over the full rotational group due to the axial symmetry of the Hamiltonian relative to the field. This, together with (7.19), yields... 

Here n is an operator of molecular axis orientation. In the classical description, it is just a unitary vector, directed along the rotator axis. Angle a sets the declination of the rotator from the liquid cage axis. Now a random variable, which is conserved for the fixed form of the cell and varies with its hopping transformation, is a joint set of vectors e, V, where V = VU...VL,.... Since the former is determined by a break of the symmetry and the latter by the distance between the molecule and its environment, they are assumed to vary independently. This means that in addition to (7.17), we have... 

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