Cross product, of vectors

The vector product or "cross" product (term coined by Gibbs53) is defined only in three-dimensional space The vector product, or cross product, of vectors a and b is a vector v, whose magnitude is a b sin y, where y is the angle between a and b, and whose direction is perpendicular to both a and b, and whose orientation is such that a, b, and v form a right-handed system ... 

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

If you know about the cross product of vectors, this follows directly from A = a X b = afc sin0 = x y2 — y x2.) This combination of variables has the form of a determinant, written as... 

The tensor (cross) product of a tensor with a vector is found as follows... 

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

The time dependence of the magnetization vector, M(t), is thus related to the cross-product of M and B. Keep in mind also that the magnetic field can be time-dependent. We have replaced B0 by B to indicate that the magnetic field can consist... 

A second question arises for those who understand the importance of dimensional analysis, a subject that is treated briefly in Appendix II. If A and B are both vector quantities with, say, dimensions of length, how can their cross product result in a vector C, presumably with dimensions of length The answer is hidden in the homogeneous equations developed above [Eqs. (IS) to (20)]. The constant a was set equal to unity. However, in this case it has the dimension of reciprocal length. In other words, C = aABsirtd is the length of the vector C. In general, a vector such as C which represents the cross product of two ordinary vectors is an areal vector with different symmetry properties from those of A and B. 

The first term is characterized by a scalar, 7, and it is the dominant term. Be aware of a convention disagreement in the definition of this term instead of -27, some authors write -7, or 7, or 27, and a mistake in sign definition will turn the whole scheme of spin levels upside down (see below). The second and third term are induced by anisotropic spin-orbit coupling, and their weight is predicted to be of order Ag/ge and (Ag/ge)2, respectively (Moriya 1960), when Ag is the (anisotropic) deviation from the free electron -value. The D in the second term has nothing to do with the familiar axial zero-field splitting parameter D, but it is a vector parameter, and the x means take the cross product (or vector product) an alternative way of writing is the determinant form... 

Two vectors commonly represented in terms of cross products are the angular momentum of a particle about some point, equal to the cross product of the momentum vector of the particle and the radius vector from the origin to the particle and torque, equal to the cross product of the force vector and the vector representing the lever arm. 

The curl of a vector function (or rotation with symbol rot ) is a vector that is formally the cross product of the operator and the vector. For a vector V,... 

Classically the angular momentum M about a point in space is the cross product of the distance vector r and the linear momentum p ... 

The vector product (cross product) of two vectors produces a vector... 

Thus the scalar product of vectors A and B in two-dimensional space is equal to the sum of the products of their components with no cross terms (e.g., AxBy). This result is actually only a special case of the general rule in p-dimensional space ... 

The metric coefficient in the theory of gravitation [110] is locally diagonal, but in order to develop a metric for vacuum electromagnetism, the antisymmetry of the field must be considered. The electromagnetic field tensor on the U(l) level is an angular momentum tensor in four dimensions, made up of rotation and boost generators of the Poincare group. An ordinary axial vector in three-dimensional space can always be expressed as the sum of cross-products of unit vectors... 

There is also a relation between polar unit vectors, boost generators, and electric fields. An electric field is a polar vector, and unlike the magnetic field, cannot be put into matrix form as in Eq. (724). The cross-product of two polar unit vectors is however an axial vector k, which, in the circular basis, is e<3>. In spacetime, the axial vector k becomes a 4 x 4 matrix related directly to the infinitesimal rotation generator /3) of the Poincare group. A rotation generator is therefore the result of a classical commutation of two matrices that play the role of polar vectors. These matrices are boost generators. In spacetime, it is therefore... 

The research problems examined so far have involved process or mixture variables only, hence n = p or n = q (see Figure 8.11). These are special cases of the general problem where both types of variables are present, namely n= p + q. The problem where both process and mixture variables are taken into consideration was formulated for the first time by Scheffe [4], The mixture-process factor space is a cross product of the mixture and process factor spaces. Each vector x = a, x3,. .., xq, xq+l,. .., xq+p=n consists of q coordinates, for which the conditions described in Equation 8.10, Equation 8.11, and Equation 8.22 hold. The remaining coordinates represent the values of process variables. The usual practice is to use transformed or coded values of the process variables rather then the natural ones (see Equation 8.8 and Section 8.2.2.1). 

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. 

Any dipolar magnetic field pattern is symmetric with respect to rotations around a particular axis. Hence, it is customary to describe the magnetic dipole moment that creates such a field as a vector with a direction along that axis. The SI units of magnetic moments are thus A m. From Eq. 8.2, the torque experienced by the magnetic moment in the external field is given by the cross product of the magnetic moment and... 

The classical expression for the magnitude of the angular momentum of a rotating particle having mass m and describing a circular path of radius r with speed v is mvr. Since it is also necessary to characterize the plane and the direction of the rotation, the orbital angular momentum is a vector equal to the cross product of the position vector r and the linear momentum p ... 

Conversely, the vector (or cross) product of the same two vectors (vixv ) is defined as a vector V3 in the direction perpendicular to the plane of Vi and V2, whose magnitude is equal to the product of the absolute values of the two vectors and the sine of the angle a between them, or... 

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.

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