Vector algebra length

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. 

Note that, although we treat x and f in the same manner, they are in fact different types of vectors. The vectors x and v are the standard vectors for position and velocity used in continuum mechanics. The internal-coordinate vector on the other hand, is a generalized vector of length N in the sense of linear algebra. 

Algebraic description of symmetry operations is based on the following simple notion. Consider a point in a three-dimensional coordinate system with any (not necessarily orthogonal) basis, which has coordinates x, y, z. This point can be conveniently represented by the coordinates of the end of the vector, which begins in the origin of the coordinates 0, 0, 0 and ends at x,y, z. Thus, one only needs to specify the coordinates of the end of this vector in order to fully characterize the location of the point. Any symmetrical transformation of the point, therefore, can be described by the change in either or both the orientation and the length of this vector. 

To obtain an algebraic (as well as geometric) way of representing vectors, we set up Cartesian coordinates in space. We draw a vector of unit length directed along the positive X axis and call it i. (No connection with i = /. ) Unit vectors in the positive y and z directions are called j and k (Fig. 5.2). To represent any vector A in terms of the three unit vectors, we first slide A so that its tail is at the origin, preserving its direction... 

A complex is called short, if it is not longer than two. A mechanism is a second order mechanism, if all the reactant complexes are short and if there exists at least one of length two. A set of elementary reactions is said to be independent if there is no way of expressing any of the elementary reaction vectors as a linear combination of the others. In the opposite case the elementary reactions are said to be dependent. From this definition it is clear that the number of independent elementary reactions is the number of independent columns of y. But this number is called in linear algebra the rank of y rank(y). This number is usually denoted by S and is considered as the dimension of the stoichiometric space, i.e. the dimension of the linear... 

Since projections are scalar values, projection of the resulting vector will be found as an algebraic sum of the projections of the individual vectors. The maximum value of the projection of the vector L is = M, where = (i+(2> whereas the shortest length of vector L will be if projections have opposite signs, L = Thereby, when adding the angular momen-... 

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