Direction in 3-D space

A space vector represents a quantity having a magnitude and direction in 3-D space reference to a co-ordinate system is not necessary. A complex number is a time vector in a 2-D plane and always referred to a Cartesian co-ordinate system. 

Therefore, we can project the AB vector in 3-D space onto 2-D space by using a projection onto the (x, z) plane, resulting in a point on a vector (on the 2-D (x, z) plane) the vector being 6.33 units in length and having an X-direction angle equal to 71.57° (as in Figure 13-4). 

Figure 13-5 By projecting a vector in (x, y, z) space onto a plane in (x, z) space, and by an x-directional rotation of 71.57° in the (x, -) plane, we have the reduction of a point on a vector in 3-D space to a point on a vector in 1-D space.

Fig. 5.1 Some fiber orientation distributions and corresponding second-order orientation tensor components a fully aligned in the 1-direction b random in the 1-2 plane c random in 3-D space...

FIGURE 11.21 The 90% boundary plots for the real forms of p and d wavefunctions. The specific label on the p or d orbital depends on the direction the orbital takes in 3-D space. 

A real object can possess more than one of the same type of symmetry element. For example, benzene has several rotational axes, as shown in Figure 13.6a. In real objects, the proper axis of rotation that has the largest n (an n-fold axis) is called the principal axis. It is conventional to consider the principal axis to be the z-axis in 3-D space. In the identification of axes of rotation, both directions of rotation (clockwise and counterclockwise) need to be considered independently, so that a rotation of 90° in a clockwise direction is not the same as a 90° rotation in the counterclockwise direction. 

The model protein is used to search the crystal space until an approximate location is found. This is, in a simplistic way, analogous to the child s game of blocks of differing shapes and matching holes. Classical molecular replacement does this in two steps. The first step is a rotation search. Simplistically, the orientation of a molecule can be described by the vectors between the points in the molecule this is known as a Patterson function or map. The vector lengths and directions will be unique to a given orientation, and will be independent of physical location. The rotation search tries to match the vectors of the search model to the vectors of the unknown protein. Once the proper orientation is determined, the second step, the translational search, can be carried out. The translation search moves the properly oriented model through all the 3-D space until it finds the proper hole to fit in. 

Correct. And a 3-D space cuts a 4-D hyperspace into two pieces. In general, an -dimensional space cuts an n + l)-dimensional space in half. For our discussion. I ll refer to the two regions of hyperspace, separated by the 3-D space, as located in the upsilon and delta directions. The words upsilon and delta can be used more or less like the words up and down. To cement the terms in your mind, think of Fleaven as lying a mile in the upsilon direction, and Hell residing a mile in the delta direction. Of course, you can t see either living in our 3-D world. ... 

You throw one of the balls to Sally. The curvature of our 3-D universe would be in the direction of the fourth dimension. Our straight lines would actually be curved, but in a direction unknown to us. This would be similar to a creature living on the two-space surface of a sphere. Lines that appeared straight to him would actually be curved. Parallel lines could actually intersect, just as longitude lines (which seem parallel at the equator) intersect at the poles. This curvature could be hard to detect if his, or our, universe were large compared to the local curvature. In other words, only if the radius of the hypersphere (whose hypersurface forms our 3-D space) were very small, could we notice it. ... 

In such a small universe, if you run in a straight line, you d return to your starting point very quickly. In any direction you looked, you d see yourself (Fig. 4.2a). You pause dramatically before launching into a more intriguing line of thought. The idea that our 3-D space is the surface of the hypersphere is seriously considered by many responsible scientists. This idea suggests another, even wilder possibility. ... 

Therefore, the section shown in Figure 15, and the pore network in 3-D from which it arises, are both absolutely defined in a quantitative way. Inasmuch as the 3-D networks are felt to be a realistic representation of random pore spaces, it is feasible to compute directly several important macroscopic properties for the FCC powder particles. Amongst these properties are permeability and effective difflisivity, so that diffusion and reaction calculations relevant to gas-oil cracking in the FCC particles can be directly undertaken. Also important in this respect are calculations of deactivation due to coke laydown within the particles. It is also possible that the pore networks could be used to deduce strength and abrasion resistance of the particles. 

The phase appears, when the system makes a trip" in configurational space. We may make the problem of the Berry phase more familiar by taking an example from everyday life. Let us take a 3-D space. Put your arm down against your body with the thumb directed forward. During the operations described below, do not move the thumb with respect to your arm. Now stretch your arm horizontally sideways, rotate it to your front, and then put down along your body. Note that now your thumb is not directed toward your front anymore, but toward your body. When your arm went back, the thumb made a rotation of 90. ... 

Two different types of optical switches have been developed. One is called 3-d, for three-dimensional switching, where optical signals are directed through a 3-d space. The other is called 2-d, because the optical signals are directed in the two-dimensional space above a wafer surface. For performance reasons, 3-d switches are used in larger arrays ( > 32),... 

The value of the coefficient of turbulent diffusion, D, depends upon the air change rate in the ventilated space and the method of air supply. Studies by Posokhin show that approximate D values for locations outside supply air jets is equal to 0.025 m-/s. Air disturbance caused by operator or robot movement results in an increase in the D value of at least two times. Studies by Zhivov et al. showed that the D value is affected by the velocity and direction of cross-drafts against the hood face, and the presence of an operator e.g., for a cross-draft directed along the hood face with velocity u = 0.5 m/s with D = 0.15 m-/s (with the presence of an operator), an increase to = 1.0 m/s results in D = 0.3 m-/s. 

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