Rotation vector

As a first application, consider the case of a single particle with spin quantum number S. The spin functions will then transform according to the IRREPs of the 3D rotational group SO(3), where a is the rotational vector, written in the operator form as [36]... 

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 voltage can also be pictured as a rotating vector with its own amphtude and frequency. Both current and potential can be represented as having real (observed) and imaginary (not observed) components. 

This represents a shaking force at a frequency equal to rotor speed. For foundation design, a value of 5 to 10 times the residual unbalance or A to 1 times rotor weight at operating speed would be a reasonable design value. The direction of the force is perpendicular to the shaft, and operates as a rotating vector which can be centered between the bearings. 

When a sinusoidally varying stress is applied to a material it can be represented by a rotating vector as shown in Fig. 2.53. Thus the stress at any moment in time is given by... 

Since the potential and current are sinusoidal, the impedance has a magnitude and a phase, which can be represented as a vector. A sinusoidal potential or current can be pictured as a rotating vector. For standard AC current, the rotation is at a constant angular velocity of 60 Hz. 

PAHs introduced in Section 34.1. A PCA applied on the transpose of this data matrix yields abstract chromatograms which are not the pure elution profiles. These PCs are not simple as they show several minima and/or maxima coinciding with the positions of the pure elution profiles (see Fig. 34.6). By a varimax rotation it is possible to transform these PCs into vectors with a larger simplicity (grouped variables and other variables near to zero). When the chromatographic resolution is fairly good, these simple vectors coincide with the pure factors, here the elution profiles of the species in the mixture (see Fig. 34.9). Several variants of the varimax rotation, which differ in the way the rotated vectors are normalized, have been reviewed by Forina et al. [2]. 

The components of the translation and rotation vectors are given as Tx> Ty, T and RX Ry, Rz, respectively. The components of the polarizability tensor appear as linear combinations such as axx + (xyy> etc, that have the symmetry of the indicated irreducible representation. 

FIGURE 16.5 The rotation of polatized light represented as a rotated vector. 

The graphic demonstration thereof is self-explanatory. The linear(resultant) vector defines a fluctuating electromagnetic field. Each of the rotating vectors describes a helix around the time axis. 

While trying to resolve which sources are present in the data, one starts with an initial guess of the elemental composition of the source material. This concentration profile is then used as the test vector, b, in equation 21. From the rotation vector and b, a predicted vector, b, can be calculated. The error observed between the original test vector b and the predicted test vector b gives an indication as to whether the test vector is a reasonable representation of a factor. 

Figure 3.10 Transformation properties of translational and rotational vectors.

The C(0) part has the same effect on Rm as on a translational vector along the x axis. However the effect of a reflection on Rx is opposite to its effect on a translational vector along the x axis. Thus the matrices describing the effect of 5(0) on the rotation vectors are the negatives of the matrices describing the effect of 5(0) on the translational vectors. The character of any 5(0) for P is the negative of its character in P. 

Since rrot is reducible to the direct sum of three one-dimensional representations, a rotation vector along any one of the three coordinate axes forms... 

Note that the matrix U simply rotates vectors that are already in the plane defined by u to a standard coordinate system. Figure 7.30 shows the data points of a color circle transformed... 

Rot(axis,theta,c) rotates vector c around axis by theta c=Rot(axis,theta,c-a)/sqrt(lambda)+[0.5,0.5,0.5] Outputlmage[x,y]=c... 

Let us for simplicity discuss a triatomic molecule, for example H2O, with fi perpendicular to the plane defined by the three atoms. In that case, the photon will mainly excite molecules that are perpendicularly aligned to the Eo vector, i.e., that lie in a plane perpendicular to Eo- If the dissociation time is small compared to the rotational period of the parent molecule, the rotational vector of OH will be preferentially directed parallel to the laboratory 2-axis because the recoil of H and OH proceeds in-plane. This would lead to a distribution in the projection quantum number mj which is strongly peaked near mj j. For a parallel transition, on the other hand, we would expect the opposite situation, i.e, j would be aligned perpendicularly to the 2-axis and P(mj) would peak near mj - 0. 

Figure 2.6 Diagram to describe the angular momentum associated with a rotating vector.

In the above equation (5.2) where X is the original vector represented by N components, Y is the new, rotated vector represented by the same number of components, while A is the rotation matrix. In the case of the Fourier transformation the rotation matrix A (eq. 5.2) is written as a matrix W ... 

Figure 4-6 illustrates a vector rotated by an angle a in the xy plane. The coordinates of the rotated vector are related to the coordinates of the original vector in the following way (r is the length of the vector, (3 is an auxiliary angle shown in Figure 4-6, and the rotation is anticlockwise) ... 

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