Vectorial angle

Cubic point groups have four threefold axes (3 or 3) that mutually intersect at angles of 109.47°. They correspond to the four body diagonals of a cube (directions x+y+z, -x+y-z, -x-y+z and x-y-z, added vectorially). In the directions x, y, and z there are axes 4, 4 or 2, and there can be reflection planes perpendicular to them. In the six directions x+y, x-y, x+z,. .. twofold axes and reflection planes may be present. The sequence of the reference directions in the Hermann-Mauguin symbols is z, x+y+z, x+y. The occurrence of a 3 in the second position of the symbol (direction x+y+z) gives evidence of a cubic point group. See Fig. 3.8. 

The velocity of the plug in the downchannel (z) and tangential (0) directions can be vectorially visualized in Fig. 5.3. Notice that the volumetric flow rate, 0, as well as all of the plug velocities are dependent upon the unknown solids forwarding angle q>. 

Contact angle measurement is probably the most common method for solid surface tension measurement in condensed state. Young [71] described the equilibrium at three-phase boundary in terms of the vectorial sum, as shown in Fig. 3, resulting in the following equation of equilibrium forces balance... 

The wetting of the solid by a liquid is a contact angle phenomenon as shown in Figure 9.3. For a drop of liquid in contact with a solid, the vectorial balance of the surface tensions of the solid-vapor, solid-liquid, and liquid-vapor gives the well-known Young s equation [8, 9] ... 

This valency angle is no new idea in organic chemistry. The essential fact is that this angle is now obtained directly for the first time by means of vector combination of the moments. Hitherto, however, it has only been possible to carry out these calculations for a few substances. Hence the fact is of importance that (as I shall show by means of one or two examples) the depolarization of scattered light and the Kerr effect enable us to prove the existence of such angles and to calculate their value, even when free rotation, which gives rise to errors in the vectorial combination, is present. 

As will be explained in Section 8.3.3, the reduction spectrum of rank-three tensors contains, in the general case, one (pseudo)scalar, three vectors, two (pseudo)devia-tors, and one septor [13]. Here, we only have to deal with the vectorial parts. They transform as a vector. Hence, the scalar product of the dipole moment vector with a vector part of the first hyperpolarizability tensor can be written as the simple product of two scalars, when we designate 9 as the angle between the molecular axis (z-axis) and the dipole moment axis (Eq. (14)). 

These expressions relate the components of the complex compliance to the amplitudes of both the perturbation and the response as well as to the out-of-phase angle 5. The vectorial scheme of the perturbation and response is shown in Figure 6.3b. 

Shape Coordinates. The gradients of any explicitly defined curvilinear shape coordinate can be found by the methods of geometric algebra. The results given in Table I are useful in this context. We solve the gradients of a bond angle 0paT by direct vectorial differentiation as a simple example ... 

Oscillatory analysis is used to determine the phase angle (8) and the amplitude of the strain (y0) that results from the application of an oscillatory stress of amplitude (og). From these, it is possible to vectorially resolve the response into elastic and viscous components. 

In polyatomic molecules the dipoles associated with each bond add vectorially to give a resultant that depends on the bond angles. In highly symmetrical molecules such as BF3, CF4,... 

Figure 9.1 Vectorial equilibrium for a drop of a liquid resting on a solid surface to balance three forces, namely, the interfacial tensions, between solid and liquid, ySo that between solid and vapor, yv, and that between liquid and vapor, jw, resulting in Young s equation (ySv = Kl + )fycos0), where 9 is the contact angle. The down component of the vector forces (-yLVsin 9) is also shown.

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