Vector orthogonal components

The equations of Table 2-5 can be used to define orthogonal components of motion in space, and these components are then combined vectorally to give the complete motion of the particle or point in question. 

The application of principal components regression (PCR) to multivariate calibration introduces a new element, viz. data compression through the construction of a small set of new orthogonal components or factors. Henceforth, we will mainly use the term factor rather than component in order to avoid confusion with the chemical components of a mixture. The factors play an intermediary role as regressors in the calibration process. In PCR the factors are obtained as the principal components (PCs) from a principal component analysis (PC A) of the predictor data, i.e. the calibration spectra S (nxp). In Chapters 17 and 31 we saw that any data matrix can be decomposed ( factored ) into a product of (object) score vectors T(nxr) and (variable) loadings P(pxr). The number of columns in T and P is equal to the rank r of the matrix S, usually the smaller of n or p. It is customary and advisable to do this factoring on the data after columncentering. This allows one to write the mean-centered spectra Sq as ... 

An n-dimensional vector is considered to have n orthogonal components and the vector is defined by specifying all of these as an array. There are two conventions to define either a column vector, like... 

Fig. 17. Imaging unsteady-state and turbulent flow of water within a 29-rnni diameter pipe. Three orthogonal component velocity images acquired at increasing Re of (a) 1250, (b) 1700, (c) 2500, (d) 3300, fe) 4200, and (1) 5000 are shown. The color scale identifies the magnitude of the z-velocity, and the flow velocity in the plane of the image (x-y) is shown by the vectors on each image. The vector scale bar on each image corresponds to Icrns . Reprinted from reference (47), with permission from Elsevier, Copyright (2004).

The vectors whose components are the characters of two different irreducible representations are orthogonal, that is,... 

Equation (13) describes the orthogonality of the columns of the character table. It states that vectors with components /CkJg x ( f) in an lVr-dimensional space are orthonormal. Since these vectors may be chosen in Nc ways (one from each of the Nc classes),... 

Figure 1.5 Components of the spin polarization vector P of ejected photoelectrons. The direction of the photoelectron is given by the polar and azimuthal angles and O (see Fig. 1.4). For an ensemble of electrons emitted in this direction, the polarization vector P then points in a certain direction in space, and one possibility for representing this vector using three orthogonal components is shown in the figure Plong in the direction of the photoelectron and P,ransX and P,ranS both perpendicular to this direction (for the definition and measurement of these components see Section 9.2.1).

As demonstrated by the example of section 1.2.1, interaction of the electric field with anisotropic materials can cause its orthogonal components to have dissimilar phases and amplitudes. These properties of the electric vector describe the state of polarization of the electric vector. Since the electric vector lies in the plane perpendicular to the axis of propagation, a convenient description for this purpose is the two-component vector ... 

This problem was treated in section 1.6 of Chapter 1, where the Fresnel coefficients for reflected and refracted light were calculated and presented in equations (1.74) to (1.77). The problem being treated is pictured in Figure 1.4, and it is convenient to represent the electric vector as a Jones vector having orthogonal components that are either parallel... 

The expression (1.75) is simplified if we note that acts only in the subspace of wave vectors orthogonal to K / is purely transverse. Thus, let us define for each wave vector its transverse component d by... 

Another entity that we shall need belongs to the realm of intrinsic geometry geodesic curvature. Consider a surface x, a point P on x and a curve on x passing through P. The curvature vector of at P joins P to the centre of curvature of This curvature vector may be decomposed into mutually orthogonal components. These components are given by projection of the... 

For intermediate angles, the fraction of intensity transmitted is given by the square of the co-sine of the angle (Malus law). This is easily understood as the electric field vector can be decomposed in two orthogonal components, one along the polarization axis that is transmitted, and one along the perpendicular direction that is absorbed. Since intensity is proportional to the square of the electric field amplitude, the Malus law follows. 

The Vector Surface Harmonic functions may be divided into two orthogonal components which indicate their 0 and dependence ... 

A polarizer is an optical element that attenuates the orthogonal components of the electric field vector of an electromagnetic wave unevenly. Let us denote the corresponding attenuation coefficients as pe and pq, and consider first the... 

When the electromagnetic radiation is incident at a boundary between two phases, part of the beam is reflected and part of it is transmitted into the second medium. Figure 9.1 shows the plane of incidence at the boundary formed by a transparent and an absorbing phase. The plane of incidence is the plane that contains the incident and reflected beams. Figure 9.1 shows that the incident beam can be envisaged as composed of two linearly polarized orthogonal components. These components, called p- and s-polarized components, are characterized by the directions of the electric field vectors in the plane of incidence and perpendicular to it, respectively. When the electric field of the radiation is located only in the plane of incidence, the beam is referred to as p-polarized. On the other hand, when the electric field of the radiation is located only in the plane perpendicular to the plane of incidence the beam is said to be s-polarized. 

As previously, the set of 5 X 5 matrices is a reducible representation of the group C v- In this case the matrices are diagonal and therefore in reduced form. We may therefore immediately write down the irreducible components by taking sets of equivalent diagonal elements. Thus, the irreducible representations 1, 1, —1, —1, 1, —1, 1, —1 each appear once, and 1, 1, 1, 1) appears three times. We are therefore able to express the vector G in terms of the orthogonal vectors in the space of the group elements. This procedure is extraordinarily useful in physical problems in much the same manner that it is useful to know the orthogonal components of an ordinary vector property in Cartesian space. 

The center of gravity of the molecule is placed at the origin of a Cartesian coordinate system. The molecule is oriented in this coordinate system so that the principal axes of inertia are along the coordinate axes. The vector radius R, of each mass point of the chain molecule (see Figure 4-13) can be resolved into the three orthogonal components (Ri)i, (Ri)2, and (Ri)3, where... 

The same procedure is applied to finding the next straight-line reaction path (or characteristic vector) in a system of more than three components. But it is important, as explained above, to start from a composition defined by a vector orthogonal to the first two orthogonal characteristic vectors. Otherwise, the reaction path, near equilibrium, will again be tangent to the straight-line reaction path already determined. 

Fig. 2. A simple biomechanical analysis looking at the resolution of a single axial compressive force vector into orthogonal components at the L3-4, L4-5, and L5-S1 levels. The analysis demonstrates the increased anterior shear forces at the lower lumbar levels, particularly the lumbosacral level. The analysis is overly simplistic in that it does not include the effect of the posterior musculature which may counteract these large anterior shear forces...

The electric field (polarization) vector can be divided into two orthogonal components, named s-polarized and p-polarized light (see Section 2.6). Recall that for the former the polarization vector is perpendicular to the plane of incidence (the one defined by the surface normal and the direction of incidence), and for the latter it is parallel to the plane of incidence. The key point is to investigate whether the angular-dependent photolysis cross-section, for a given wavelength and polarization, tracks the metal... 

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