Scalar products

The scalar product (dot product) of two vectors produces a scalar. 

The scalar product between two vectors is a scalar quantity represented as V -V and is defined by the following equation  

Finally, the scalar product can be also written as a matrix product  

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Taking advantage of the synnnetry of the crystal structure, one can list the positions of surface atoms within a certain distance from the projectile. The atoms are sorted in ascending order of the scalar product of the interatomic vector from the atom to the projectile with the unit velocity vector of the projectile. If the collision partner has larger impact parameter than a predefined maximum impact parameter discarded. If a... 

As the scalar product of two vectors is related to the cosine of the angle included by these vectors by Eq. (4), a frequently used similarity measure is the cosine coefficient (Eq. (5)). 

Some of the common manipulations that are performed with vectors include the scalar product, vector product and scalar triple product, which we will illustrate using vectors ri, T2 and r3 that are defined in a rectangular Cartesian coordinate system ... 

We can now proceed to the generation of conformations. First, random values are assigne to all the interatomic distances between the upper and lower bounds to give a trial distam matrix. This distance matrix is now subjected to a process called embedding, in which tl distance space representation of the conformation is converted to a set of atomic Cartesic coordinates by performing a series of matrix operations. We calculate the metric matrix, each of whose elements (i, j) is equal to the scalar product of the vectors from the orig to atoms i and j ... 

For functions of one or more variable (we denote the variables collectively as x), the generalization of the vector scalar product is... 

Note that relations (1.91) and (1.92) mean linearity of the duality mapping I and its inverse I in Hilbert spaces due to the linearity of the scalar product. 

On the other hand, the duality mapping I is defined by the scalar product in the Hilbert space V. Assume that the operator A is self-conjugate. Then we can define the scalar product in V as follows ... 

This means that A is the duality mapping connected with the introduced scalar product ( , - )a- Then the variational inequality (1.126) can be rewritten in the form... 

The brackets ( , ) denote the scalar product in L Q). The aim of further reasonings is a proof of the following statement. 

In fact, by the second Korn inequality this scalar product induces a norm which is equivalent to the norm given by (5.3). Hence, because (/, p) = 0 for all p G R fl), the identity (5.29) actually holds for every u G Therefore, the equilibrium equations... 

Two approaches to this equation have been employed. (/) The scalar product is formed between the differential vector equation of motion and the vector velocity and the resulting equation is integrated (1). This is the most rigorous approach and for laminar flow yields an expHcit equation for AF in terms of the velocity gradients within the system. (2) The overall energy balance is manipulated by asserting that the local irreversible dissipation of energy is measured by the difference ... 

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