Vector cross product

The torque is given by the vector cross product of the vectors pi and Hq. 

Following usual conventions, repeated indices indicate summation and fy denotes df/dXj. The permutation S5mibol is used to present the vector cross product in indicial notation. Due to the anisotropic nature, traction and body couples can exist, and thus the angular momentum equation must be considered. For purely viscous fluids this equation says simply that the deviatoric stresses are symmetric. 

It is helpful to exemplify the basis by calculating the vector cross-product in detail and comparing it with the Cartesian counterpart. This procedure shows that the ((1),(2),(3)) and Cartesian representations are equivalent when correctly worked out. 

This can be extended straightforwardly to angular momentum operators and infinitesimal magnetic field generators. Therefore, a commutator such as Eq. (818) is equivalent to a vector cross-product. If we write Bm> as the scalar magnitude of magnetic flux density, the commutator (818) becomes the vector cross-product... 

Well-known realizations of the generators of this Lie algebra are given by the three components of the orbital angular momentum vector L = r x p, the three components of the spin S = a realized in terms of the Pauli spin matrices (Schiff, 1968), or the total one-electron angular momentum J = L + S. The components of each of these vector operators satisfy the defining commutation relations Eq. (4) if we use atomic units. We should also note that the vector cross-product example mentioned earlier also satisfies Eq. (4) if we define E = iey, j = 1, 2, 3. 

It was mentioned earlier that a number of special purpose routines, which do not appear in the VPLIB index, have been developed for use in structural chemistry. The most frequent requirements encountered in this area are those concerned with molecular geometry and, more specifically, with the calculation of interatomic distances, angles and torsion angles. These geometric quantities are best evaluated by vector algebra and this will always involve the calculation of vector components, lengths, direction cosines, vector cross products and vector dot products. Attention should therefore be directed at the best possible way of implementing the calculations described in the latter list on the MVP-9500. 

In Bloch s original treatment of NMR,23 he postulated a set of phenomenological equations that accounted successfully for the behavior of the macroscopic magnetization M in the presence of an rf field. These relations are based on Eq. 2.41, where M replaces X, and B is any magnetic field—static (B0) or rotating (B,). By expanding the vector cross product, we can write a separate equation for the time derivative of each component of AT ... 

The vector (cross) product of vectors P and R is a vector Q orthogonal to both P and R of magnitude given by... 

Figure 1.39. Vector (cross) product of two vectors. The orientation of V3 is determined using the right-hand rule thumb of the right hand is aligned with Vj, index finger with V2, then V3 is aligned with the middle finger. Tails of all vectors face the middle of the palm.

In an isotropic medium, such as a liquid, the polarization is given by tire vector cross product of the electric fields... 

Variation principle 18, 154, 222 VB (valence bond) model 94 Vector 4 Vector docking 57 Vector potential 294 Vector space 220 Vector, cross product 6 Vector, dot product 5 Vectors, orthogonal 6 Velocity dipole operator 193 Velocity relaxation 253... 

A special direction, known as the zone axis, is the one that is common to two planes h kih and 2 2/2-The directions [hikM and [h2k2l2 are the normals to the two planes and the zone axis [UVW is then given by the vector cross-product. The zone axis has particular significance in electron microscopy because it represents the direction of the incident electron beam with respect to the sample. 

A measure for the alignment can eithCT be defined by means of the vector dot product or by the vector cross product For small alignment angles, i.e. almost parallel shafts, the second is more sensitive. 

The alignment vector can be defined as the normalized vector cross product a X b... 

A condition involving these three vectors may be formulated using vector products, specific to three dimensions. Orientation of a plane can be defined by its normal vector n (a vector perpendicular to the plane). Since r(C) and v(C) span a two-dimensional plane, a vector cross product involving r(C) and v(C) may be applied to find n. Hence, n = v(C) X r(C). If the vector dr(C)v(C) is to lie in the plane... 

The normal vector n(C) may also be computed from the vector cross product of v(C) and r(C) ... 

Since operates specifically on vectors in (due to the use of the vector cross product), r(C) and (C-C ) are vectors composed of only Cg, Cg, and Cj—this is unlike the r(C) and v(C) vectors used by the standard approach using Lie brackets, or the computation of the determinant function A(C) for critical CSTR concentrations, which are methods that both require vectors of... 

The easiest way to do this is to construct a set of orthogonal axes using three tracking targets and a series of vector cross products [Eqs. (5.28) to (5.30)]. The resulting orthogonal axes are illustrated in Fig. 5.12. These axes will be referred to as a local coordinate system. 

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