Addition and Subtraction of Vectors

The sum of a and b is given by the vector c, which is found in the following series of steps  

The directed line segment OQ represents the vector c, defined as the sum of a and b. Furthermore, as OQ = OP + PQ = OR + RQ, it follows that c = a + b = b + a, from which we see that addition is commutative in other words, a displacement OR followed by RQ clearly leads to Uie same final point as a displacement OP followed by PQ. 

Since OR and PQ are equivalent, and represent the same vector b, we can use a triangle to summarize the relationship between a, b and the 

The subtraction of two vectors can be thought of as the addition of two vectors that differ in their sign. If we think of this in terms of displacements in space, then the first vector corresponds to a displacement from point P to point Q, for example, whereas a second identical vector with opposite sign will direct us back to point P from point Q  

It follows that subtraction of two vectors, a and b, is equivalent to adding the vectors a and — A, and so we can define vector subtraction in a general sense as  

---

Figure 3.6-1. Addition and subtraction of vectors (a) addition of vectors, B + C (b) subtraction of vectors, B — C.

Addition and subtraction of matrices are performed element-wise the matrices must have the same size (see Figure A.2.2). Multiplication of a vector or a matrix with a scalar is also performed element-wise in the case of a vector the resulting vector has the same direction but a different length. Vectors a and b = —a have the same length but reverse direction. 

The addition and subtraction of matrices, which now encompass vectors as well, is directly the addition and subtraction of the elements, analogous to the rules for scalars. 

Any kind of operation on a vector, including addition and subtraction, can be somewhat laborious when working with its graphical representation. However, by referring the vectors to a common set of unit vectors, termed base vectors, we can reduce the manipulations of vectors to algebraic operations. 

When the two molecules are on the other hand not translationally equivalent, as is the case in the unit cell of anthracene - Fig. 6.8, left - then both excited states have nonvanishing transition moments. Their magnitudes and directions can be computed with the aid of Eq. (6.7) for dipole transitions by addition or subtraction of the individual vectors. This is demonstrated schematically in Fig. 6.8. 

The following important rules of tensor analysis should be mentioned Addition and subtraction is only defined for tensors of the same rank and of the same transformation properties (co-/contravariance). For example, adding a matrix and a vector is not valid. Multiplication (also called tensor contraction) is only defined for pairs of indices, where one index is co- and the other is contravariant. As another example, odxi is a valid tensor contraction, but x x is not. 

It should be noted that in the above presentation of the combination of vectors by addition or subtraction, no reference has been made to their components, although this concept was introduced in the beginning of this chapter. It is, however, particularly useful in the definition of the product of vectors and can be further developed with the use of unit vectors. In the Cartesian system employed in Fig. 1 the unit vectors can be defined as shown in Fig. 4. 

FIGURE 3.22 The total dipole moment of a molecule is obtained by vector addition of its bond dipoles. This operation is performed by adding the arrows when they lie pointing in the same direction, and subtracting the arrows if they lie pointing in different directions, (a) CO2. (b) OCS. (c) H2O. (d) CCI4. 

The algebra of multi-way arrays is described in a field of mathematics called tensor analysis, which is an extension and generalization of matrix algebra. A zero-order tensor is a scalar a first-order tensor is a vector a second-order tensor is a matrix a third-order tensor is a three-way array a fourth-order tensor is a four-way array and so on. The notions of addition, subtraction and multiplication of matrices can be generalized to multi-way arrays. This is shown in the following sections [Borisenko Tarapov 1968, Budiansky 1974],... 

Note that if only the positive vector components were allowed, then they would not form an Abelian group (no neutral element), and on top of that, their addition (which might mean a subtraction of components, because a, /8 could be negative) eould produce vectors with nonpositive components. Thus, the veetors with all positive eomponents do not form a vector space. 

A vector is not necessarily referred to a Cartesian coordinate system, but, for example, to a neighbor vector with respect to magnitude and direction. Rules about the addition, subtraction, and multiplication (dot and cross products) of two vectors are a part of vector algebra. 

Vectors. Velocity, force, momentum, and acceleration are considered vectors since they have magnitude and direction. They are regarded as first-order tensors and are written in boldface letters in this text, such as v for velocity. The addition of the two vectors B -i- C by parallelogram construction and the subtraction of two vectors B — C is shown in Fig. 3.6-1. The vector B is represented by its three projectionsB, By, and Bj on the x, y, and z axes and... 

Since it is necessary to represent the various quantities by vectors and matrices, the operations for the MND that correspond to operations using the univariate (simple) Normal distribution must be matrix operations. Discussion of matrix operations is beyond the scope of this column, but for now it suffices to note that the simple arithmetic operations of addition, subtraction, multiplication, and division all have their matrix counterparts. In addition, certain matrix operations exist which do not have counterparts in simple arithmetic. The beauty of the scheme is that many manipulations of data using matrix operations can be done using the same formalism as for simple arithmetic, since when they are expressed in matrix notation, they follow corresponding rules. However, there is one major exception to this the commutative rule, whereby for simple arithmetic ... 

A vector can be thought of as a point in -dimensional space, although the graphical representation of such a point, when the dimension of the vector is greater than 3, is not feasible. The general rules for matrix addition, subtraction, and multiplication described in Section A.2 apply also to vectors. 

The ASC differs from the conventional scalar computer in that it is a pipeline computer with a full set of hardware vector instructions in addition to the standard scalar instructions. The vector hardware includes arithmetic operations such as add, subtract, multiply, divide, vector dot product, as well as vector instructions for shifting, logical operations, comparisons, format conversions, normalization, merge, order, search, peak pick, select, replace, MIN, and MAX. Although an ASC may have one to four pipes, the configuration described below will be that of the two pipe machine at NRL. 

To return to the crystallographic experiment itself the addition of such a heavy atom must result in a measurable change in the structure factors F /. If we denote the structure factors in the absence of the heavy atom as FP (the protein Fs) and those in its presence as FPH (the protein-and-heavy-atom Fs), the difference FPH— Fp is Fh, the contribution of the heavy atom(s) alone. As the structure factors are complex, the subtraction must be represented in an Argand diagram as a vector difference (Figure 19). 

---