Collinear vectors

Figure 2. Orthogonal decomposition of a three-dimensional Hilbert space case of two collinear vectors in the two-dimensional subspace.

Any two vectors in a plane are linearly independent as long as they are noncollinear. Two collinear vectors are linearly dependent. However, three noncollinear vectors in a plane must be linearly dependent. Thus the plane satisfies the requirements of a two-dimensional vector space. It should be emphasized that there is an infinite number of possible sets of basis vectors in any space. 

Whatever the exact definition of this coupling frequency, the net result of the internal coupling is the equality between the energy-per-entity of the products times entity number of each energy sub-variety (for the sake of generality, dot products are used, although these state variables are collinear vectors) ... 

In such a case, the two vectors being collinear, do not form a basis of the S subspace and, consequently, do not entirely define the subspace they belong to. 

Another example of the potential utility of polarized edge spectra for structure determination is found for [MoO J2" (28). This molecule has C2V symmetry and the C2 axes of all of the molecules in the unit cell are collinear. Thus, when the crystal is oriented with the polarization parallel to the S-S interatomic vector, the polarization is perpendicular to the Mo-0 bonds and nearly parallel to the Mo-S bonds. Similarly, the crystal can be oriented with the polarization perpendicular to the Mo-S bonds and nearly parallel to the Mo-0 bonds. For both orientations, excellent agreement was obtained with SCF-X a calculations of the edge structure (j ). 

As easily seen, the vector PA is always parallel to the surface plane and the sin-face-parallel component of the vector Ps vanishes for the structures having collinear projections of molecular axes onto the surface plane (

[Pg.68]

Spin I> 0 nuclei possess a magnetic dipole or dipole moment, n, which arises from a spinning, charged particle. Nuclei that have a nonzero spin will also have a magnetic moment, and the direction of that magnetic moment is collinear with the angular momentum vector associated with the nucleus. This can be expressed as... 

Combining this expression with (3.153), and using the continuity equation for (U), yields S"p = 0. Note that local isotropy does not exclude the possibility that scalar gradient vectors are collinear as is often the case when l = Vp. 

Nevertheless, if (3.78) is known to be violated, a further issue is to find the variable that is primarily responsible for the violation. The ratio of the absolute value of the correction to the corresponding standard deviation provides some information but may be misleading (ref. 31). The analysis proposed by Almdsy and Sztand (ref. 32) is based on geometric ideas. If exactly one observation is corrupted by gross error then the corresponding column of matrix W and the vector f of equation errors are nearly collinear. Useful measures of collinearity are 7 = cos, where is the... 

An additional advantage of PCR is that the score vectors are orthogonal (PCA provides orthogonal scores), so that the collinearity problems found in ILS are avoided and the resulting parameter estimates a are stable. Note also that this model also maintains the form described in eqn (3.3). This can be seen by starting with the original ILS model [eqn (3.26)] and multiplying by... 

Figure 6.7. The ML6 octahedral complex. Unit vectors a,n,n are oriented parallel to the orthonormal axes x, y, z, which have their origin at M and lie along OX, OY, OZ. The three C2 axes that are collinear with the C4 axes are along x, y, z, and the second set of six C2 axes that bisect the angles between x and y, y and z, and z and x are designated C2. The symmetry planes that contain these C2 axes are ad planes because they bisect the angles between C2 axes that are normal to one of the three C4 principal axes.

It was shown that for all values of a the condensate function consists of a singlet (SC) and a triplet (TC) components. Even in the case of a homogenous magnetization (a = 0), in addition to the SC, the TC with the zero projection onto the z axis arises. In this case, both the SC and the TC decay in the F layers over a short distance given by = y/Dp/J. If the magnetization vectors M are not collinear a / 0,7r/2, all projections of the TC appear, in particular, those with non-zero projection on the 2-axis. In this case, the TC... 

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