Scalar and Vector Products

In this volume, we will often use the scalar and vector products of two vectors. Thus, we will start with a review of vector algebra. The scalar product of vectors a and b is defined as 3i = aJbx + a y + aibz = abcosd. (1-1) 

6 is the angle between a and b as shown in Fig. l-l(a). The scalar product is also called the inner product, which means the degree of the overlap between a and b. 

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The complex product aj)pears to be made up of two terms, not too unlike the scalar and vector products, from which it seems to differ only because of a sign convention. 

Using scalar and vector products among the eigenvectors (direction cosines), demonstrate that the principal coordinates remain orthogonal. 

Matrices and Determinants Revisited Alternative Routes to Determining Scalar and Vector Products... 

Reciprocal quantities may be calculated using formulae which are derived from multiple scalar and vector products ... 

Using rules for scalar and vector products, this equation can be rewritten due to symmetry of the stress tensor ... 

Most workers have preferred to use spherical atom-atom functions in simulations, because they are easier to handle computationally. However the formulae for handling anisotropic atom-atom functions are available[5 ], and explicit expressions have been given in terms of scalar and vector products of the vectors describing relative positions and orientations[55]. These vectors are already used in most simulation techniques. Representations of charge distributions in terms of arrays of point charges require more interaction centres (and an increase in computational time proportional to the square of the number of centres) and do not address the description of anisotropy in the repulsion. The necessity for using anisotropic atom-atom repulsion potentials is illustrated by the success of a recent model of this kind used to... 

Here we have introduced Q, the tensor of second electric moments, which is related to the quadrupole moment of the electron distribution, and the scalar and vector product operators for two tensors, O and 0. The expansion may, of course, be carried to... 

Each operator thus reduces to a spin-free part involving a scalar product of momentum operators, a spin-orbit term involving the spin of both electrons, and a spin-spin interaction term. Because of the projection operators, no further simplifications can be made in general in these expressions apart from rearrangement of the scalar and vector products into other forms. 

It has been assumed in deriving these expressions that the momentum operators are always operating on the basis functions, and not on any of the operators involving Vij, with which they will therefore commute. This is because the reduction using (15.2) requires the rearrangement of the order of the scalar and vector products in the intermediate stages. Nevertheless, since p x ry commutes with any scalar function of Vij, the first term in each of the above expressions is in general correct even if p is not taken to operate on a basis function. 

In the smectic C phase, the layer normal z and the director n represent the only two natural directions which we can define. That is, z and n are the only natural vectors in the medium. Using these we now want to construct an order parameter which is a vector, is chiral, has C2 symmetry, leads to a second order A -C transition and is proportional to 6 for small values of 6. We can do this by combining their scalar and vector products, multiplying one with the other. 

Scalar and vector products, together with the usual ideas of gradient, diyergence and curl, will be required in Cartesian notation. Briefly, the scalar product of the vectors a = (ai, U2, as) and b = (6i, 62,63) is deflned by... 

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