Cartesian coordinate system origin

Concurrent engineering Engineering analysis Cartesian coordinate system Origin Plane... 

A particle moving with momentum p at a position r relative to some coordinate origin has so-called orbital angular momentum equal to L = r x p. The three components of this angular momentum vector in a cartesian coordinate system located at the origin mentioned above are given in terms of the cartesian coordinates of r and p as follows ... 

If a vector it is a function of a single scalar quantity s, the curve traced as a function of s by its terminus, with respect to a fixed origin, can be represented as shown in Fig. 8. Within the interval As the vector AR = R2 - Ri is in the direction of the secant to the curve, which approaches the tangent in the limit as As - 0. Tins argument corresponds to that presented in Section 2.3 and illustrated in Fig. 4 of that section. In terms of unit vectors in a Cartesian coordinate system... 

Also included in this work is a comparison of the direction of the dipole moments for each compound using two different measurement techniques. The first technique involves comparing the vector components against a standard Cartesian coordinate system, where the center of mass of the molecular model rests at the origin and the inertial axes of the molecule are oriented along X, Y, and Z axes, respectively. This represents the dipole in coordinate space, as shown in Figure 1. 

The factor r1 enters because the Cartesian spherical harmonics clmp are defined in terms of the direction cosines in a Cartesian coordinate system. The expressions for clmp are listed in appendix D. As an example, the c2mp functions have the form 3z2 — 1, xz, yz, (x2 — y2)/2 and xy, where x, y and z are the direction cosines of the radial vector from the origin to a point in space. 

Problem 5-1. Visualize or construct graphs in two dimensions, of the geometric interpretations of the 2-vector (3, 4) with respect to two or more Cartesian coordinate systems. These coordinate systems may differ with respect to the position of the origin and directions of the axes however, the axes must in every case be perpendicular to one another. Keep the scale the same. 

Any displacement of the atoms in a molecule can be described in terms of displacements in Cartesian coordinate systems centered at each atom. Thus, for water we may imagine three sets of axes, originating at the equilibrium positions of each of the three atoms, as shown in Figure 7.1. 

FIGURE 9.2 Sketch of far-field detection of CARS waves generated in the focal volume. The figure shows the yz cross-section of the excitation volume. O is the origin of the coordinate system and also the center of the focal volume and O is a far-field point on the optical axis. The coordinates of the points in the near-field are represented by (x,y,z) and those of the points in the far-field by (X,Y,Z) in cartesian coordinate system. The outgoing arrows around the near-field points represent the CARS waves generated in the focal volume. The incoming arrows at the far-field points represent the contributions from the individual points in the focal volume. 

Let us consider a vector in ordinary three-dimensional space. We can specify the length and direction of this vector in the following way. We arrange to have one end of the vector lie at the origin of a Cartesian coordinate system. The other end is then at a point which may be specified by its three Cartesian coordinates, x, y, z. In fact, these three coordinates completely specify the vector itself provided it is understood that one end of the vector is at the origin of the coordinate system. We can then write these three coordinates as a column matrix, in this case one with three rows, x y z, and say that the matrix represents the vector in question. 

Symmetry axes of a Cartesian coordinate system with the metal ion at the origin, as shown in Fig. 

Figure 2.1 The Cartesian coordinate system used to represent the points (3.3) and (-3,-3) in the plane defined in terms of coordinates referenced to the origin (0,0)...

Solution Select a Cartesian coordinate system with the origin on the axis of rotation and the x-axis coinciding with the line connecting the origin and the initial location of the particle, as shown in Fig. E3.2. The velocity of gas is given by... 

A particle is located along the x-axis in a Cartesian coordinate system, 1 unit from the origin. Find its position in spherical coordinates. 

Figure 8.11. Cartesian coordinate system for describing the position vectors of the particles (electrons and nuclei) in a molecule. 0(X, Y, Z) is the laboratory-fixed frame of arbitrary origin, and c.m. is the centre-of-mass in the molecule-fixed frame. For the purposes of illustration four particles are indicated, but for most molecular systems there will be many more than four.

A column matrix consists of only one column. Column matrices are used to represent vectors. A vector is characterized by its length and direction. A vector in three-dimensional space is shown in Figure 4-4. If one end of the vector is at the origin of the Cartesian coordinate system, then the three coordinates of its other end fully describe the vector. These three Cartesian coordinates can be written as a column matrix ... 

First, an appropriate basis set has to be found. Considering that a molecule has 37V degrees of motional freedom, a system of 37V so-called Cartesian displacement vectors is a convenient choice. A set of such vectors is shown in Figure 5-5 for the water molecule. A separate Cartesian coordinate system is attached to each atom of the molecule, with the atoms at the origin. The orientation of the axes is the same in each system. Any displacement of the atoms can be expressed by a vector, and in turn this vector can be expressed as the vector sum of the Cartesian displacement vectors. 

Taking the B atom as the origin of a Cartesian coordinate system, along any axis there is a dimer unit consisting of a B atom and an O atom at each lattice point. Consequently, each dimer unit has a bonding and antibonding tt level, given from Table 4.2, as ... 

The trick, of course, is calculating Q. Fortunately, we don t have to know the absolute values of S and Q, we re more interested in how these quantities change as a result of some process, like stretching a chain. This allows us to use probability distributions, which sounds rather awful, but for a lot of problems is fairly straightforward. For example, if we take a single chain and fix one of its ends at the origin of a Cartesian coordinate system and let the other end be at a position (Figure 13-49), then... 

A molecule-fixed Cartesian coordinate system is oriented such that its origin coincides with the center of mass and the axes coincide with the main inertial axes of the molecule. A rotation axis can thus be identified by a Cartesian axis, for example C (z). The main axis is usually defined as the 2 axis. Planes are identified by the axes they contain a(xy) or tr.y. 

In the first part of this chapter we studied the radial vibrations of a solid or hollow sphere. This problem was considered an extension to the dynamic situation of the quasi-static problem of the response of a viscoelastic sphere under a step input in pressure. Let us consider now the simple case of a transverse harmonic excitation in which separation of variables can be used to solve the motion equation. Let us assume a slab of a viscoelastic material between two parallel rigid plates separated by a distance h, in which a sinusoidal motion is imposed on the lower plate. In this case we deal with a transverse wave, and the viscoelastic modulus to be used is, of course, the shear modulus. As shown in Figure 16.7, let us consider a Cartesian coordinate system associated with the material, with its X2 axis perpendicular to the shearing plane, its xx axis parallel to the direction of the shearing displacement, and its origin in the center of the lower plate. Under steady-state conditions, each part of the viscoelastic slab will undergo an oscillatory motion with a displacement i(x2, t) in the direction of the Xx axis whose amplitude depends on the distance from the origin X2-... 

Consider a set A and a (possibly approximate) symmetry element R, where the associated symmetry operator R leaves at least one point of the convex hull C of set A invariant. We assume that a reference point c g C, a fixed point of R, and a local Cartesian coordinate system of origin c are specified, where the coordinate axes are oriented according to the usual conventions with respect to the symmetry operator R. For example, if R is a Cy rotation axis, then the z axis of the local Cartesian system is chosen to coincide with this Cj axis, whereas if R is a reflection plane, then the z axis may be chosen perpendicular to this plane. 

Let us draw a sphere, Cji, with the center at the origin of the Cartesian coordinate system and of a radi is R, big enough that domain Q belongs completely to the ball Oil bounded by the sphere Cg Q C Og (Figure 8-1). Applying inequality (8.112) to the difference field inside this ball, Og, we have... 

We introduce now a domain Vr, bounded by a sphere dVa of a radius / , with its center at the origin of some Cartesian coordinate system, x, y, z. Integrating both sides of equation (9.65) over the domain Vr, and applying Gauss s theorem, we find ... 

---