Components of a vector

We can use this matrix to project onto the component of a vector in the v(i) direction. For the example we have been considering, if we form the projector onto the v(l) vector, we obtain... 

As indicated above, if the components of a vector X in n-dimensional space are real, the vector can be written as a column matrix with n rows. Similarly, a second vector Y in the same space can be written as a column matrix of the... 

The value of the dot product is a measure of the coalignment of two vectors and is independent of the coordinate system. The dot product therefore is a true scalar, the simplest invariant which can be formed from the two vectors. It provides a useful form for expressing many physical properties the work done in moving a body equals the dot product of the force and the displacement the electrical energy density in space is proportional to the dot product of electrical intensity and electrical displacement quantum mechanical observations are dot products of an operator and a state vector the invariants of special relativity are the dot products of four-vectors. The invariants involving a set of quantities may be used to establish if these quantities are the components of a vector. For instance, if AiBi forms an invariant and Bi are the components of a vector, then Az must be the components of another vector. 

If Tz is interpreted as the z-component of a vector, the three-dimensional circulation integral per unit area defines the vector curl v. 

Any set of quantities that transform according to this prescription are known as the covariant components of a vector, and represented by subscripted indices3. 

The quantities /" are the contravariant components of a vector in the coordinate system X. They give an actual vector only when multiplied by the unit vector e = hvev. If the unit vectors along the coordinate lines have a scale inverse to the coordinate, e = ev/hv so that... 

It is worth remarking that a gas sensor array is a mere mathematical construction where the sensor outputs are arranged as components of a vector. Arrays can also be utilized to investigate the properties of chemical sensors, or even better, the peculiar behaviour of a sensor as a component of an array. In this chapter, the more common sensor array methodologies are critically reviewed, including the most general steps of a multivariate data analysis. The application of such methods to the study of sensor properties is also illustrated through a practical example. 

For simulation purposes n random numbers from a normal distribution N(p,a2) can be generated as components of a vector x by... 

In this section, quantities are expressed with respect to rectangular Cartesian co-ordinate axes, Ox, 0x2, Ox, except where otherwise stated. The components of a vector V are denoted Vi, V2 and F3 with respect to these axes. 

Vector projection operators act by cancelhng out all components of a vector except the one it is designed to select. The decomposition of a function in an analogous way requires expression of the function as a sum of components each of a proper symmetry species. For example, it is possible to write any function of three variables as a sum of components that are symmetric or antisymmetric with respect to inversion ... 

After projection of a component of a vector, reapplication of the same projection operator has no effect. 

Physically, this means that when you project onto the components of a vector in these three directions, you don t lose any of the vector. This happens because our vectors are orthogonal and complete. The completeness relation means that any vector in this three-dimensional space can be written in terms of v(l), v(2), and v(3) (i.e., we can use v(l),v(2),v(3) as anew set of bases instead of ei,e2,e3). 

This expression must be independent of the choice of the co-ordinate system. In line with the general usage to look upon the 6 independent elements of the symmetric strain (stress) tensor as components of a vector, we may elucidate this independence of co-ordinate system by writing eq. (2) as the scalar product of the vectors [fijrjn yy< Exyyj2., vj /2, and [/l2, fl2 3 > 2 /2, fafa-J2, faP -y/2]. The... 

In the case of the flux of mass, the result is the normal component of pua. But for the flux of momentum and energy, in general the flux density is not the normal component of a vector or tensor function of (t, x), since it will depend on the extended shapes of if and Y. But in the case of short-range forces and slowly varying p, ua, E, it can be shown to have this form with sufficient approximation. Thus one is led to the familiar pressure tensor and heat flow vector Qa, both as functions of (t, x). It is to be emphasized that the general expression of these quantities involves not only expected values of products of momenta (or velocities), but the effect of intermolecular forces. 

Colorimetry [1.17] - [1.19]. The principles of colorimetry are based on the fact that all color stimuli can be simulated by additively mixing only three selected color stimuli (trichromatic principle). A color stimulus can, however, also be produced by mixing the spectral colors. Thus, it has a spectral distribution, which in the case of nonluminous, perceived colors is called the spectral reflectance q (2). After defining three reference stimuli, the trichromatic principle allows a three-dimensional color space to be built up in which the color coordinates (tristimulus values) can be interpreted as components of a vector (CIE system for standards, see Table 1, Colorimetry CIE = Commission Internationale de l Eclairage). For uncolored illumination the three CIE tristimulus values depend on the spectral reflectance as follows ... 

This formidable expression can be written more compactly. Regard Ff(i, t) as the components of a vector P(r, t) and define the matrices... 

This means that in the set of matrices constituting any one irreducible representation any set of corresponding matrix elements, one from each matrix, behaves as the components of a vector in /i-dimensional space such that all these vectors are mutually orthogonal, and each is normalized so that the square of its length equals hllh This-interpretation of 4.3-1 will perhaps be more obvious if we take 4.3-1 apart into three simpler equations, each of which is contained within it. We shall omit the explicit designation of complex conjugates for simplicity, but it should be remembered that they must be used... 

Any set of algebraic functions or vectors may serve as the basis for a representation of a group. In order to use them for a basis, we consider them to be the components of a vector and then determine the matrices which show how that vector is transformed by each symmetry operation. The resulting matrices, naturally, constitute a representation of the group. We have previously used the coordinates jc, y, and z as a basis for representations of groups C2r (page 78) and T (page 74). In the present case it will be easily seen that the matrices for one operation in each of the three classes are as follows ... 

Figure E.9 Spatial orientation and z components of a vector with l = 2.

For example, the components of a vector r in configuration space may be represented by the column matrix xyz). 

To solve the single-particle problem it is convenient to introduce a new representation, where the coefficients ca in the expansion (1) are the components of a vector wave function (we assume here that all states a are numerated by integers)... 

Polarizabilities have been determined and compared for a number of azaindole compounds. The data are summarized in Table 33 <90JPC1755>. Overall, the results show that the different structural isomers have very similar values for the polarizability (a) and the second hyperpolarizability (y) while there are dramatic differences in the first hyperpolarizability (/ ). This may result from the fact that a and y are scalar properties while / is a component of a vector. 

In the first place, we shall find that the four quantities Ty px0y py0y pz0 must be constant at all points of space, for equilibrium. By comparison with Eq. (2.4) of Chap. IV, the formula for the Maxwell distribution of velocities, we see that T must be identified with the temperature, which must not vary from point to point in thermal equilibrium. The quantities pxo, pyo, p 0 are the components of a vector representing the mean momentum of all the molecules. If they are zero, the distribution (2.15) agrees exactly with Eq. (2.4) of Chap. IV. If they are not zero, however, Eq. (2.15) represents the distribution of velocities in a gas with a certain velocity of mass motion, of components pxo/my pyQ/my pzo/m. The quantities px — pxo, etc., represent components of momentum relative to this momentum of mass motion, and the relative distribution of velocities is as... 

References Components of a Vector, Line Segment, Pythagoras Theorem, Translation, Trigonometry. 

We generally denote scalars by lowercase Greek letters (e.g., P), column vectors by boldface lowercase Roman letters (e.g., x), and matrices by capital italic Roman letters (e.g., H). A superscriptT denotes a vector or matrix transpose. Thus xT is a row vector, xTy is an inner product, and AT is the transpose of the matrix A. Unless stated otherwise, all vectors belong to R , the u-dimen-sional vector space. Components of a vector are typically written as italic letters with subscripts (e.g., xux2,.. . , ). The standard basis vectors in R" are the n vectors ei,e2,. . . , e , where e has the entry 1 in the th component and 0 in all others. Often, the associated vector norm is the standard Euclidean norm, j 2, defined as... 

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