Unit vectors direction cosines

The radial frequency co of a periodic function is positive or negative, depending on the direction of the rotation of the unit vector (see Fig. 40.5). co is positive in the counter-clockwise direction and negative in the clockwise direction. From Fig. 40.5a one can see that the amplitudes (A jp) of a sine at a negative frequency, -co, with an amplitude. A, are opposite to the values of a sine function at a positive frequency, co, i.e. = Asin(-cor) = -Asin(co/) = This is a property of an antisymmetric function. A cosine function is a symmetric function because A -Acos(-co/) = Acos(cor) = A. (Fig. 40.5b). Thus, positive as well as negative... 

We can simplify this expression by decomposing the vectors along the three axes, with unit vectors i, j and k and introducing the direction cosines of the bond (1,2), namely cosri2, cosyi2 and coszi2- The z-components of V12 and V34, cancel, while V23 is directed along x. This yields the equation... 

When V has unit length (i.e., a unit vector n) the direction cosines are the components of the unit vector ... 

The components of n are the direction cosines of the unit vector (Eq. A. 112) in the same coordinate system as the components of T. The system of three equations represented by Eq. A.137 contains only two independent equations. The solution for n indicates a direction, but can be arbitrary in magnitude. For n to be a unit vector, the solution must also satisfy... 

Referring to Fig. A.2, assume that the principal coordinates align with z, r, and O. The unit vectors (direction cosines) just determined correspond with the row of the transformation matrix N. Thus, if the principal stress tensor is... 

Let the Cartesian coordinate axes x y z have the same origin as the xyz axes. The x y z set is obtainable from the xyz set by rotation, reflection, or inversion, or some combination of these operations. (If the x y z set is left handed while the xyz set is right handed, we must perform a reflection or inversion as well as a rotation to generate the x y z axes from the xyz axes.) Let the vector r have coordinates (x,y,z) and (x, y, z ) in the two coordinate systems. If i is a vector of unit length along the x axis, then (1.55) gives r i —Let be the direction cosines of the x" axis... 

When used as the dispersion formula for the phonons and polaritons in orthorhombic crystals, the symbols in Eq. (11.22) have the following meaning r)= 1,2,3 designates the three directions of the principal orthogonal axes. sv are the direction cosines of the normalized wave vector s = k/k with respect to the three principal axes of the crystal. If the unit vectors in the directions of these three principal axes are designated eue2,e3, one can write... 

We can now use the results that we have obtained as a guide to the general problem of waves propagated in an arbitrary direction. To describe the direction of propagation, imagine a unit vector along the wave normal. Let the x, y, and z components of this unit vector be Z, m, n. These quantities are often called direction cosines, for it is obvious that they are equal respectively to the cosines of the angles between the direction of the wave normal and the x, y, z axes. Then in... 

The components of the vector u (m, Uy, u/) must not be confused with direction indices, which are normally enclosed in brackets instead of carets. If the rotation axis is speci-hed in terms of direction indices, one hrst has to convert these indices into direction cosines in order to use Eq. 1.7. The direchon cosines are the scalar components of a unit vector expressed as a linear combinahon of the Cartesian basis vectors i,j, and k. The value of each component is equal to the cosine of the angle formed by the unit vector with the respective Cartesian basis vector. For example, the body diagonal of a cube of unit length has direction indices [1 1 1]. The body diagonal runs from the origin with Cartesian coordinates (xi, yi, Zj) = (0, 0, 0) to the opposite comer of the cube with Cartesian coordinates (x2, y2, Z2) = (T T 1)- The direction cosines, referred to our Cartesian basis vectors, are given by the equations ... 

Fortunately, if the rotation axis corresponds to a Cartesian coordinate axis, considerable simphfications ensue. First of aU, the direction indices of the Cartesian coordinate axes are thex-axis = [10 0], they-axis =[01 0], and the z-axis = [001]. These indices are identical with the i,j, and k unit vectors that are co-directional with the x, y, and z axes, respectively. For example, if the rotation axis is the z-axis, described by the direction indices [mi w] = [00 1], these indices are numerically equivalent to the direction cosines cos a = m = 0, cos /3 = m = 0, and cos y=u = 1, since cos a + cos /3 -I-cos 7=1- Now, in this book the standard convention is followed that a clockwise rotation by a vector in a fixed coordinate system makes a negative angle and a counterclockwise rotation, a positive angle. Therefore, with a counterclockwise rotation about the vector u = 0, 0, 1), the z-axis, Eq. 1.7 reduces to (see Example 1.2 below) ... 

The components of the electric field vector E now need to be found along the three mutually perpendicular axes (x, y, z), which are simply the projections of the vector on the axes for this, direction cosines are used. The vector E is directed along [111], which is the body diagonal. The body diagonal runs from the origin with Cartesian coordinates (xt, Yu Zi) = (0, 0, 0) to the opposite corner of the unit cell, in this case with Cartesian coordinates (X2, Y2. Z2) = (a, b, c) = (5.10, 6.25, 2.40). So, the direction cosines are given by ... 

With the construction of a unit vector along E using the direction cosines as its components along thex, y, and z directions, the vector E can be expressed as ... 

Wc need finally the atomic matrix elements <0, a H i, /i>, the form of which has been given by Slater and Koster (1954). The vector r,. is written (/x + my -t- i z) f/, with x, y, and z unit vectors along the cube axes that is, /, m, and II are the direction cosines of the vector from the left state to the right state. Then the matrix elements arc written as an E, and the states represented by their angular form are written as subscripts, the first for the left state (a), and the second for the right state (/i) for example, the symbol represents <0, a // i, (i),... 

Thus, the rate of energy transfer depends on the square of the dot product between the acceptor dipole (transition dipole) Pa and the field Eo of the donor dipole (transition dipole), po (Eq. 11 and Fig. 3b). For any chosen locations and orientations of the donor and acceptor, the value of involves the cosine of the angle between the unit vectors Po and Pa (i.e., po Pa) as well as the cosine of the angles between f and Pa (i.e., r Pa) and between f and Po (i.e., f -Pd)- Therefore, for any constant selected angle between the donor and acceptor dipoles (that is, constant Po Pa), the value of will depend on the position in space where the acceptor dipole is relative to the donor. The strength of the field of the donor molecule for any particular constant values of Qo, 0a and Po Pa changes with the distance r as 1/r, that is, for any particular direction of r relative to po As illustrated in Fig. 3a, for a particular angle between the orientations of the donor and acceptor dipoles (Po Pa),... 

Let 0 be the origin and OE1( OEa.. . OE, unit lengths along the axes of the (uq, w2. . . u r)-co-ordinate system. Let P0, Px, Pg.. . be the points of intersection of the path, confined to the unit cube, with the (/—l)-dimensional surfaces bounding the unit cube, which intersect in OEu OEj.. . OE,. Let P and 0 be identical. Since the direction cosines are incommensurable none of these points P coincide they have at least one limit point in each of the bounding surfaces perpendicular to the axes. In each of these (/—l)-dimensional surfaces, there is therefore an infinite number of vectors PmPm+n, whose magnitudes are less than a given number 8. 

The second factor in the expression of Eq. 8.(>8 is a unit vector along the directions of either bond 2 (+ sign) or bond 3 (— sign). The cosine of the angle between the bonds is the scalar product of these two unit vector s thus... 

In the above, ai, 2 and 3 are called the directional cosines and are the components of the normal unit vector, i.e., the line of unit length perpendicular to a surface at point P. They are usually written as ... 

We use (4.24a) and (4.24b) and the direction cosine displacement coordinates of Walmsley and Pople (1964) to describe the application to a linear molecule as given by Brith, Ron, and Schnepp (1969). Equivalent treatments have been given by Cahill (1968) and by Richardson and Nixon (1968). The components A,(// ) of the unit vector A(// ) are the instantaneous direction cosine of the molecular axis. Part of the components of the matrix of the transformation from molecular (primed) to crystal (unprimed coordinates (4.24b) can be identified with the Aj, i.e.. 

An alternate method of deriving the vectors S( is illustrated in the case of the bending coordinate as follows. The scalar product of unit vectors directed outwards from the central atom gives the cosine of the angle ... 

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