Vector dot product

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

The force / and displacement s are vector quantities, and equation (2.5) indicates that the vector dot product of the two gives a scalar quantity. The result of this operation is equation (2.6)... 

We can use the generalized expression for hybrids to find the relation between hybridization indices and angle between two hybrids xi and X2 (Equations A1.10). Since the expressions in braces are equivalent to ordinary three-dimensional vectors of unit length, the three-dimensional vector dot product, found by... 

A closely similar complex circular basis has been described by Silver [112] for three-dimensional space. This space forms the internal gauge space in 0(3) electrodynamics, as argued already. In the complex circular basis, the unit vector dot product is... 

Figure 8.9 A sampling playback oscillator using high order interpolation. Every output sample is a vector dot product of fFinput samples and one of the filter coefficient vectors, stored re-ordered from the original prototype filter. The fractional phase address selects the filter coefficient vector used.

Where B0 is a vector representing the external magnetic field, J is the coupling constant, Ia and 7b are the vectors representing the nuclear magnets of Ha and Hb and the products are vector dot products. The first part represents the energy of interaction of the spins with the B0 field and the second part represents the energy of interaction of the spins with each other. Since the B0 vector has only a z component, we can write the vector products as... 

The ASC differs from the conventional scalar computer in that it is a pipeline computer with a full set of hardware vector instructions in addition to the standard scalar instructions. The vector hardware includes arithmetic operations such as add, subtract, multiply, divide, vector dot product, as well as vector instructions for shifting, logical operations, comparisons, format conversions, normalization, merge, order, search, peak pick, select, replace, MIN, and MAX. Although an ASC may have one to four pipes, the configuration described below will be that of the two pipe machine at NRL. 

Finally, most doubly or triply subscripted array operations can execute as a single vector instruction on the ASC. To demonstrate the hardware capabilities of the ASC,the vector dot product matrix multiplication instruction, which utilizes one of the most powerful pieces of hardware on the ASC, is compared to similar code on an IBM 360/91 and the CDC 7600 and Cyber 174. Table IV lists the Fortran pattern, which is recognized by the ASC compiler and collapsed into a single vector dot product instruction, the basic instructions required and the hardware speeds obtained when executing the same matrix operations on all four machines. Since many vector instructions in a CP pipe produce one result every clock cycle (80 nanoseconds), ordinary vector multiplications and additions (together) execute at the rate of 24 million floating point operations per second (MFLOPS). For the vector dot product instruction however, each output value produced represents a multiplication and an addition. Thus, vector dot product on the ASC attains a speed of 48 million floating point operations per second. 

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. 

Equation 6.5 is just the expanded form of a vector dot product and is more useful for later computations. For N nuclei the sum includes N(N — 1) terms. For example, for a system of four nuclei there are four terms in and six terms in... 

Where E = + is the total mechanical energy. The solid dots in these equations represent vector dot product. The instability is related to... 

The most general expression for a periodic function is the plane wave, e , in which d is a parameter equal to the vector dot product k R. Hence, the wave function, ip(r), of Eq. 5.24 is of the form ... 

Another operation of importance involves the gradient vector dot product ... 

Variation principle 18, 154, 222 VB (valence bond) model 94 Vector 4 Vector docking 57 Vector potential 294 Vector space 220 Vector, cross product 6 Vector, dot product 5 Vectors, orthogonal 6 Velocity dipole operator 193 Velocity relaxation 253... 

This projected density is computationally efficient. The integral in FY has analytical form. Algorithms for the vector dot product in Eq. [49] are well known, and vector computers are particularly adept at this operation. 

The key to the distance geometry method is the metric matrix, G. Each element gj of G can be calculated by taking the vector dot product of the coordinates of atoms i and . That is. 

This is a statement of the product rule for the divergence of the vector dot product of a tensor with a vector, which is valid when the tensor is symmetric. In other words, r = r, where is the transpose of the viscous stress tensor. Synunetry of the viscous stress tensor is a controversial topic in fluid dynamics, bnt one that is invariably assumed. is short-hand notation for the scalar double-dot product of two tensors. If the viscous stress tensor is not symmetric, then r must be replaced by in the second term on the right side of the (25-29). The left side of (25-29), with a negative sign, corresponds to the rate of work done on the fluid by viscous forces. The microscopic equation of change for total energy is written in the following form ... 

The vector dot product between Ji and J2 in each volume element, divided by the current squared, is now the sensitivity of the volume element and if we multiply with the resistivity p in each volume, we will directly obtain this volume s contribution to the total measured resistance R. 

Use the vector dot product to find the obtuse angle between two diagonals of a cube. What is the chemical significance of this angle ... 

When F and x are not parallel, we replace the integrand in (2.1.1) with the component of F that is parallel to x this may be found by forming the vector dot product between the force vector and a unit vector parallel to x. For a differential change in position dx, we need only a small amount of work 5W,... 

A measure for the alignment can eithCT be defined by means of the vector dot product or by the vector cross product For small alignment angles, i.e. almost parallel shafts, the second is more sensitive. 

The equivalent of a vector dot product is defined as the integral of the product of the two functions. 

The vector dot product indicates the directional sensitivity shown in Figure 6.4. The... 

To analyze the situation with a tetrapolar electrode system in contact with, for example, a human body, we must leave our simplified models and turn to lead field theory (see Section 6.4). The total measured transfer impedance measured is the ratio of recorded voltage to injected current according to Eq. 6.39. The impedance is the sum of the impedance contributions from each small volume dv in the measured volume. In each small volume, the resistance contribution is the resistivity multiplied by the vector dot product of the space vectors (the local current density from a unit reciprocal current applied to the recording electrodes) and (the local current density from a unit current applied to the true current carrying electrodes). With disk-formed surface electrodes, the constrictional resistance increase from the proximal zone of the electrodes may reduce sensitivity considerably. A prerequisite for two-electrode methods is therefore large band electrodes with minimal current constriction. 

Both force and distance are vectors, so a more general definition would be work. = F d, where denotes the usual vector dot product. 

---