Elementary Vector Algebra

Quantities that have both magnitude and direction are called vectors. Examples of vectors in chemistry include things like the dipole moments of molecules, magnetic and electric fields, velocities and angular momenta. It is convenient to represent vectors geometrically and the simplest example is 

It has two components, (x, y) in cartesian coordinates, and these can be transformed into plane polar coordinates to give 

A position vector like p above is usually represented as starting at the origin, but it remains unchanged when moved to another position in the plane, as long as its length and direction do not change. Such an operation is used to form the sum of two vectors, A and B. By moving either A or B to start from the end point of the second vector, the same vector sum is obtained. It is easy to see how the components add up to 

The product of two vectors A B is the simple product of the magnitudes AB, only if the two vectors are parallel. 

If the angle between the vectors is a, their product, called the scalar or dot product, is defined as A B = AB cos a, which is the product of any of the two vectors in magnitude times the projection of the other (B or A ) on the direction of the first. 

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A3 = Aizk Azk Elementary vector algebra then gives... 

I assume that you are familiar with the elementary ideas of vectors and vector algebra. Thus if a point P has position vector r (I will use bold letters to denote vectors) then we can write r in terms of the unit Cartesian vectors ex, Cy and as ... 

Using elementary analytical geometry, we can easily calculate the locations of the various vertexes as the simplex moves. It is necessary to consider the coordinates of each point as components of a vector and only apply the rules of vector algebra. Thus, for example, the vector located at point P is the average of the B and N vectors ... 

In elementary algebra, a linear function of the coordinates xi of a variable vector f = (jci, JT2,..., Jc ) of the finite-dimensional vector space V = V P) is a polynomial function of the special form... 

I have assumed that the reader has no prior knowledge of concepts specific to computational chemistry, but has a working understanding of introductory quantum mechanics and elementary mathematics, especially linear algebra, vector, differential and integral calculus. The following features specific to chemistry are used in the present book without further introduction. Adequate descriptions may be found in a number of quantum chemistry textbooks (J. P. Lowe, Quantum Chemistry, Academic Press, 1993 1. N. Levine, Quantum Chemistry, Prentice Hall, 1992 P. W. Atkins, Molecular Quantum Mechanics, Oxford University Press, 1983). 

In this section, some elementary details of the complex circular basis algebra generated by ((1),(2),(3)) are given. The basis vectors are... 

There is a number of references [1-13] in which an algorithm of the kinetic models construction is described, but mainly two widely used methods are applied, namely linear algebra [1, 2, 7, 10-13] and the theory of graphs (5, 6, 8, 9]. In the most of the proposed algorithms the main attention is paid into obtaining the expression for the rate of an elementary reaction. Principally, it suffices to use the vector of a rate of an elementary reaction to determine the vector of the rate of a composite substance s formation and in such a way to describe the evolution of a chemical system s composition. In particular cases, however, the expressions for the final reactions rates are retained, since in complicated systems with a set of final reactions the knowledge of an elementary reaction rate does not mean knowledge of the final reactions rates. 

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... 

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