Subtraction, of vectors

Mechanically the subtraction of vectors is very similar to addition, with the individual components being combined. Subtracting the second vector above from the first gives... 

If besides contraction, the simplex needs to change its direction, we would have a subtraction of vectors rather than an addition ... 

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

Construction of a direction specified by four indices is carried out using a procedure similar to the one used for other crystal systems—by the subtraction of vector tail point coordinates from head point coordinates. For the four coordinate axes of Figure 3.10, we use the following designations for head and tail coordinates ... 

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. 

We subtract the mean spectrum from each measured spectrum yp and as a result, the origin of the system of axes is moved into the mean. In the above example, it is into the plane of all spectral vectors. This is called meancentring. Mean-centring is numerically superior to subtraction of one particular spectrum, e.g. the first one. The Matlab program, Main MeanCenter, m, performs mean-centring on the titration data and displays the resulting curve in such a way that we see the zero us,3-component, i.e. the fact that the origin (+) lies in the (us ,i,us >2)-plane. 

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. 

Finally, recall that subtraction of (5.2.1b) from (5.2.1a) yields, taking into account (5.2.1c) and (5.2.5a), ix = 0 (this corresponds to solenoidality of the total current density vector in more than one dimension). 

Figure 23-3 Infrared absorbance spectra of the amide regions of proteins. (A) Spectra of insulin fibrils illustrating dichroism. Solid line, electric vector parallel to fibril axis broken line, electric vector perpendicular to fibril axis. From Burke and Rougvie.24 Courtesy of Malcolm Rougvie. See also Box 29-E. (B) Fourier transform infrared (FTIR) spectra of two soluble proteins in aqueous solution obtained after subtraction of the background H20 absorption. The spectrum of myoglobin, a predominantly a-helical protein, is shown as a continuous line. That of concanavalin A, a predominantly (3-sheet containing protein, is shown as a broken line. From Haris and Chapman.14 Courtesy of Dennis Chapman.

The most commonly used PCA algorithm involves sequential determination of each principal component (or each matched pair of score and loading vectors) via an iterative least squares process, followed by subtraction of that component s contribution to the data. Each sequential PC is determined such that it explains the most remaining variance in the X-data. This process continues until the number of PCs (A) equals the number of original variables (M), at which time 100% of the variance in the data is explained. However, data compression does not really occur unless the user chooses a number of PCs that is much lower than the number of original variables (A M). This necessarily involves ignoring a small fraction of the variation in the original X-data which is contained in the PCA model residual matrix E. 

If we subtract this vector from the current color, we move local space average color back to the gray vector. This is visualized in Figure 11.6. Let C = [cr, c . cb T be the color of the input pixel. Thus, output colors can be calculated by subtracting the component of local space average color, which is perpendicular to the gray vector. 

The NIPALS algorithm uses a least squares fitting to determine the direction of the first component, i.e. so that the sum of squared deviations between the data points and their projections on the component vector is as small as possible. The first component is usually determined from the residual matrix obtained after subtraction of the average matrix X from X, i.e. E = X — X. The first component vector Pl is then used to compute the first score vector, tj. 

Rk] It is important to observe the subtraction of R >0 considered in the definition of the disturbance vector. If R >0, Kg always exists. However, if we accept that R is not necessarily positive, we can have problems making the inverse matrix necessary to calculate Kg. ... 

The intensities of the scattered neutrons on the detector are then radially averaged, if the scattering unit is an isotropic scatterer, after normalization and subtraction of solvent and background, converted to absolute differential cross sections per unit sample volume, (5E/5Q) (0, in rmits of cm and plotted on a 1-D plot as a function of the scattering vector, Q. 

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 ... 

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. 

Hint the representation of vector subtraction shown in Figure 5.7(a) may be helpful. 

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