Direction of a Vector

Two vectors along the same direction have an identical set of direction ratios and vice versa. 

---

If the magnitude and direction of a vector are known, its components are the products of the magnitude and the respective direction cosines. In the case of the velocity vector, for example, the components are... 

It is often of value to define a unit vector A in the direction of a vector x. Clearly Ar = xr/ x will be covariant and its length... 

As an example, Fig. 4.4 shows another view of the Cu surface that highlights where the plane of the surface is relative to the bulk crystal structure. The orientation of this plane can be defined by stating the direction of a vector normal to the plane.1 From Fig. 4.4, you can see that one valid choice for this vector would be [0,0,1]. Another valid choice would be [0,0,2], or [0,0,7], and so on, since all of these vectors are parallel. 

Except for this change, we find hyb(l ) in the same way as before. We note, however, that this time the direction of a vector may be reversed as the result of a symmetry operation and in such a case there will be a contribution of — 1 to the character of that operation. Furthermore, we immediately see that in carrying out the different symmetry operations, no vector perpendicular to the molecular plane is ever interchanged with one in the molecular plane and vice versa. This implies two things the representation rhyb is at once in a partially reduced form (the matrices are already in block form, each consisting of two blocks) and the vectors perpendicular to the molecular plane on their own form a basis for a reducible representation of (which we will call rby ) and the vectors in the molecular plane on their own also form a basis for a reducible representation of 9 (which we will call iy.T) necessarily... 

Direction cosines, which can be used to define the direction of a vector in an orthogonal coordinate system, play an essential role in accomplishing coordinate transformations. As illustrated in Fig. A. 1, there is a vector V oriented in a (z,r,9) coordinate system. Because our concern here is only the direction of the vector, the physical dimensions are sufficiently small so that the curvature in the 9 coordinate is not seen (i.e., the coordinate system... 

It is also sometimes convenient to specify only the direction of a vector. This is done by introducing unit vectors, which are defined to have length one. Unit vectors are signified by a caret ( ) instead of an arrow. Thus v = c v = sv. 

The general equation of property conservation. For a phase defined by volume V and surface A, we consider a property which crosses the volume in the direction of a vector frequently named the transport flux ),. Inside the volume of control, the property is uniformly generated with a generation rate J. On the surface of the volume of control, a second generation of the property occurs due to the surface vector named the surface property flux Figure 3.1 illustrates this and shows a cylindrical microvolume (dV) that penetrates the volume and has a microsurface dA. 

Another example of conceptual problems is the electric dipole moment (EDM) of an electron. The EDM of an electron can be caused by two effects. One is some violation of the CP invariance in one or other way, which allows a correlation between directions of a vector (the EDM) and a pseudovector (the spin, s). All experiments have been interpreted in such a way. Meanwhile, there is another opportunity due to a possible violation of the Lorentz invariance. Such a violation can deliver a preferred frame (e.g., related either to the isotropy in CMB or to the local DM motion) and a preferred direction related to our velocity v with respect to the preferred frame. The violation could induce a certain EDM directed with this preferred direction (along v) or in a direction of [v x s]. All the experiments have been treated in the former way, while their results were considered as a model-independent limitation on the electron s EDM. 

Figures 2.5 and 2.6 show the downsizing of ft when it is, respectively, a two-component vector and a continuous function. Observe the preservation of the direction of a vector or the shape of a function during the downsizing process.

Mov ing the origin does not change the magnitude or direction of a vector, so vectors can be added, vi -i- = V2 + vi = v, by placing them head to tail as... 

The magnitude and direction of a vector do not depend on what the coordinate system is, but the components do. To dehne the Cartesian coordinate system, for example, let i be a vector that is directed along the x-axis, and... 

The proportions of p and py orbitals can be calculated using trigonometry, and the process is easier to follow if the orbitals themselves are replaced with vectors to represent their functional forms, as shown in Figure 4.9a. The direction of a vector in this diagram... 

---