Vectors

BIBLIOGRAPHY

It is well known that any point in the plane can be represented by a pair of numbers, its coordinates. If one draws a horizontal axis, labeled the x axis, and a vertical axis, labeled the y axis, then one can represent any point uniquely by giving its x coordinate and its y coordinate in order. Point (a, b ) is the point that lies directly on a vertical passing by the point marked a on the x axis and on level with the point marked b on the y axis. Naturally, a or b can be negative. Note that the order of the coordinates is important. Points represented by (a, b ) and (b, a ) are not the same (unless a = b ).

There is a complete correspondence between geometrical objects, namely, points of the plane, and purely algebraic objects, namely, ordered pair of numbers. Such algebraic objects that correspond exactly to points are called vectors. Thus a vector is a pair of any numbers (x, y ). More precisely, one should call such pairs two-dimensional vectors because the concept can be easily generalized to n-dimensional vectors, geometrical representation being of course limited to three-dimensional space; nonetheless, the illustrations herein are limited to two-dimensional space.

Economic applications of vectors are quite important and numerous. The presentation herein deals with consumption first, production and activity analysis second, and then finally, temporal processes.

Consumption structures can be easily presented using vectors. The consumption of consumer C _i consists of n different quantities or bundles of different commodities (meat, bread, beer, tea …), represented by C _i = (c _i1, c _i2, … (c _in). When for simplicity of presentation one deals with two commodities, then n = 2 and C _i = (c _i1 , c _i2), which can be represented with quantities consumed on relevant axes, as in Figure 1. The length of the vector represents the level of consumption, and its slope represents the structure of composition. Level changes of the “consumption basket” will be represented by an increase or a decrease of the vector length; changes in structure, depending on consumer preferences, revenue, and market prices, will modify the vector slope.

Vector analysis can also be useful to present problems of accumulation with heterogeneous capital goods, a technique dating as far back as Karl Marx’s analysis of “reproduction schemes” found in the second volume (1885) of Das Kapital. John von Neumann (1946) and Wassily W. Leontief (1941) have contributed important modern examples of this technique.

For an example of a production analysis, one might begin by defining an “activity,” “process,” or “production method” representing production of one commodity, say “corn,” using labor, iron, and corn (for seed needs). For one unit of labor time (one hour, one day, or labor time available in the economy), one can obtain a given amount of corn, called b ₁₁, using a ₁₁ amount of corn and a ₂₁ of iron. Consequently, activity can be represented by vector OM^→ ₁ (see Figure 2). The first coordinate of M ₁ is b ₁₁ – a ₁₁, representing “net” production of corn, that is, “gross” production of corn (b ₁₁) less intermediate consumption of corn (a ₁₁)—the seeds; the second coordinate, a ₂₁, represents the intermediate consumption of iron; vector OM^→ ₁,

representing the “net product” of activity (1), can be easily constructed by vectoral summation of, first, gross production of corn represented by vector Ob ₁₁ and second, D, vector OĀ ₁, itself sum of inputs of corn and iron.

In a similar way, let iron be produced by another activity producing b ₂₂ units (tons …) of iron using a ₂₂ units of iron, a ₁₂ units of the other commodity (corn here in this very simplified model), and one unit of labor. Vector O ₂ represents the “net product” of this activity.

One may note that OM̄ ₁ and OM̄ ₂ have just one positive component, both activities being “specialized” in the production of one and only one commodity. Such a case is called “simple production”; but one may have a different case in which one activity produces two commodities simultaneously (say, wool and milk); in such a case, named “joint production,” both coordinates of M ₃ may be positive, as indicated in Figure 2 (this vector is shown only for mathematical illustration as it is quite difficult to imagine a method producing jointly iron and corn).

When both activities are specialized, one comes to matrix representations such as:

which are square matrixes largely used in input-output systems, or Leontief models—2 × 2 in the simplified model used here, but n × n in a more general treatment. The i^ème column vector represents conditions of production of the i^ème commodity (production of b ₁₁ units of commodity i necessitates intermediate consumption of a ₁₁ units of itself and a ₂₁ units of second commodity, and so on); the i^ème line vector represents utilization of the i^ème commodity by the entire system, as intermediate consumption (a ₁₁ representing intermediate consumption of first commodity by the first production process, a ₁₂ intermediate consumption of the same commodity by the second production process, and so on).

Von Neumann models are more complicated and more general. First, there is the possibility of joint production. Second, the matrix may be rectangular because there is the possibility of production systems with un-equality between the numbers of processes (line vectors) and commodities (column vectors).

It should be noted that when economic systems can be characterized by a “square matrix,” one can define and characterize eigenvectors that have a special structure associated with matrix characteristics. Such vectors play a special role in economic analysis because they can be useful to characterize, on the one hand, special accumulation regimes—with maximum uniform growth rates—and, on the other hand, special price systems—with minimum uniform interest rates.

Vectors can also be useful in presenting and generalizing about temporal interdependencies. Simple “autoregressive” models involve the dependence of variable x on the anterior value of the same variable, for instance, x _t = f (x _t – 1), with any kind of relevant function.

Vector autoregressive (VAR) models are a generalization of such simple autoregressive models. Consider two stationary variables x _1t and x _2t; each variable depends on its own past values but also on the present and past values of the other one. VAR models are very important in contemporary econometrics. They have been introduced by Christopher A. Sims (1980) as an alternative to macro-econometric models with a Keynesian flavor.

SEE ALSO Eigen-Values and Eigen-Vectors, Perron-Frobenius Theorem: Economic Applications; Input-Output Matrix; Linear Systems; Matrix Algebra

BIBLIOGRAPHY

Leontief, Wassily W. 1941. The Structure of American Economy, 1919–1929: An Empirical Application of Equilibrium Analysis. Cambridge, MA: Harvard University Press.

Neumann, John von. 1946. A Model of General Equilibrium. Review of Economic Studies 13 (1): 1–9. (Orig. pub. 1937.)

Sims, Christopher A. 1980. Macroeconomics and Reality. Econometrica 48 (1): 1–48.

Gilbert Abraham-Frois

Vectors

A vector in the Cartesian plane is an ordered pair (a, b ) of real numbers. This is the mathematician's concise definition of a two-dimensional vector. Physicists and engineers like to develop this concept a bit more for the purpose of applying vectors in their disciplines. Thus, they like to think of the mathematician's ordered pairs as representing displacements, velocities, accelerations, forces, and the like. Since such things have magnitude and direction, they like to imagine vectors as arrows in the plane whose magnitudes are their lengths and whose directions are the directions that the arrows are pointing. The two small dark arrows shown in part (a) of the drawing below form our point of departure in the study of vectors in the plane.

These are called the unit basis vectors, or unit vectors. From them all other vectors arise. To the mathematician, they are simply the ordered pairs (1, 0) and (0, 1). To the physicist, they really are the arrows extending from the origin to the points (1, 0) and (0, 1). The horizontal one is commonly named i and the vertical one is commonly named j . To say that all other vectors in the plane arise from these two means that all vectors are linear combinations of i and j . For instance, the mathematician's vector (2, 3) is 2(1, 0) + 3(0, 1), while the physicist's arrow with horizontal displacement of 2 units and vertical displacement 3 units is more compactly written as 2i + 3j and is represented as in part (b) below.

In the vocabulary of physics, the end of the arrow with the arrowhead is called the "head" of the vector, while the end without the arrowhead is called the "tail." The tail does not necessarily have to be at the origin as in the picture. When it is, the vector is said to be in standard position, but any arrow with horizontal displacement 2 and vertical displacement 3 is regarded by physicists as 2i + 3j , as shown in part (b) of the drawing above. The 2 and 3 are called horizontal and vertical components (respectively) of the vector 2i + 3j .

By definition, if

v = (a, b ) = a i + b j

and

w = (c,d ) = c i + d j ,

then

v + w = (a + c, b + d ) = (a + c )i + (b + d )j .

This definition of addition for vectors leads to a very convenient geometrical interpretation. In part (c) of the drawing above,

v = 2i + 3j

and

w = 4i - 1j .

Then v + w = 6i + 2j .

Physicists call their convention for adding geometric vectors "head-to-tail" addition. Notice in the part (d) of the drawing above that v is in standard position and extends to the point (2, 3). If the tail of w is placed at the head of v , then the head of w ends up at (6, 2). Now if we draw the arrow from the origin to (6, 2), we have an appropriate representation of v + w in standard position. The vector v + w is usually called the resultant vector of this addition. So, in general, the resultant of the addition of two vectors will be represented geometrically as an arrow extending from the tail of the first vector in the sum to the head of the second vector. This convention is often called the parallelogram law because the resultant vector always forms the diagonal of a parallelogram in which the two addend vectors lie along adjacent sides.

Applications of Vectors

As an example from physics, consider an object being acted upon by two forces: a 30 pound (lb) force acting horizontally and a 40 lb force acting vertically. The physicist wants to know the magnitude and direction of the resultant force. The schematic diagram below represents this situation.

By the parallelogram law, the resultant of the two forces is the diagonal of the rectangle. The Pythagorean Theorem may be used to calculate that this diagonal has a length of 50, representing a 50-lb. resultant force. The inverse cosine of 30/50 is 53.13°, giving the angle to the horizontal at which the resultant force acts on the object.

Consider another example in which vectors represent velocities. An airplane is attempting to fly due east at 600 mph (miles per hour), but a 50-mph wind is blowing from the northeast at a 45° angle to the intended due east flight path. If the pilot does not take corrective action, in what direction and with what velocity will the plane fly? The drawing below is a vector representation of the situation.

The vector representing the plane's intended velocity points due east and is labeled 600 mph, while the vector representing the wind velocity points from the northeast at a 45° angle to the line heading due east. Note the head-to-tail positioning of these two vectors. Now the parallelogram law gives the actual velocity vector for the plane, the resultant vector, as the diagonal of the parallelogram with two sides formed by these two vectors. In this case, the Pythagorean Theorem may not be used, because the triangle formed by the three vectors is not a right triangle. Fortunately, some advanced trigonometry using the so-called Law of Cosines and Law of Sines can be used to determine that the plane's actual velocity relative to the ground is 566.75 mph at an angle of 3.58° south of the line representing due east. In navigation, angles are typically measured clockwise from due north, so a navigator might report that this plane was traveling at 566.75 mph on a heading, or bearing, of 93.58°.

As another example of how mathematicians and physicists use vectors, consider a point moving in the xy -coordinate plane so that it traces out some curve as the path of its motion. As the point moves along this curve, the x and y coordinates are changing as functions of time. Suppose that x = f (t ) and y = g (t ). Now the mathematician will say that the position at any time t is (f (t ), g (t )) and that the position vector for the point is R (t ) = (f (t ), g (t )) = f (t )i + g (t )j . The physicist will say that the position vector R (t ) is an arrow starting at the origin and ending with the head of the arrow at the point (f (t ), g (t )). (See the figure below.) It is now possible to define the velocity and acceleration vectors for this motion in terms of ideas from calculus, which are beyond the scope of this article.

The mathematician's definition of a vector may be extended to three or more dimensions as needed for applications in higher dimensional space. For example, in three dimensions, a vector is defined as an ordered triple of real numbers. So the vector R (t ) = (f (t ), g (t ), h (t )) = f (t )i + g (t )j + h (t )k could be a position vector that traces out a curve in three-dimensional space.

see also Flight, Measurements of; Numbers, Complex.

Stephen Robinson

Bibliography

Dolciani, Mary P., Edwin F. Beckenbach, Alfred J. Donnelly, Ray C. Jurgensen, and William Wooten. Modern Introductory Analysis. Boston: Houghton Mifflin Company, 1984.

Foerster, Paul A. Algebra and Trigonometry. Menlo Park, CA: Addison-Wesley Publishing Company, 1999.

Narins, Brigham, ed. World of Mathematics. Detroit: Gale Group, 2001.

vec·tor / ˈvektər/ • n. 1. Math. & Physics a quantity having direction as well as magnitude, esp. as determining the position of one point in space relative to another. Compare with scalar. ∎  Math. a matrix with one row or one column. ∎  a course to be taken by an aircraft. ∎  [as adj.] Comput. denoting a type of graphical representation using straight lines to construct the outlines of objects. 2. an organism, typically a biting insect or tick, that transmits a disease or parasite from one animal or plant to another. ∎  Genetics a bacteriophage or plasmid that transfers genetic material into a cell, or from one bacterium to another.• v. [tr.] (often be vectored) direct (an aircraft in flight) to a desired point.DERIVATIVES: vec·to·ri·al / vekˈtôrēəl/ adj. (in sense 1 of the noun ).vec·to·ri·al·ly / vekˈtôrēəlē/ adv. (in sense 1 of the noun ).vec·tor·i·za·tion / ˌvektərəˈzāshən/ n.vec·tor·ize / -ˌrīz/ v. (in sense 1 of the noun ).ORIGIN: mid 19th cent.: from Latin, literally ‘carrier,’ from vehere ‘convey.’

vectors

vector
1. An animal, usually an insect, that passively transmits disease-causing microorganisms from one animal or plant to another or from an animal to a human. Compare carrier.

2. (cloning vector) A vehicle used in gene cloning to insert a foreign DNA fragment into the genome of a host cell. For bacterial hosts several different types of vector are used: bacteriophages, artificial chromosomes, plasmids, and their hybrid derivatives, cosmids. The foreign DNA is spliced into the vector using specific restriction enzymes and DNA ligases to cleave the vector DNA and join the foreign DNA to the two ends created (insertional vectors). In some phage vectors, part of the viral genome is enzymically removed and replaced with the foreign DNA (replacement vectors). Retroviruses can be effective vectors for introducing recombinant DNA into mammalian cells. In plants, derivatives of the tumour-inducing (Ti) plasmid of the crown gall bacterium, Agrobacterium tumefaciens, are used as vectors. See also expression vector.

vector A one-dimensional array. In computing, vectors are widely used since the memory is essentially a vector of words. The notation for vectors is determined by the programming language. In this dictionary a bold italic lower-case letter, e.g. v, is used to denote a vector in its entirety, and the corresponding plain italic lower-case letter indexed by a subscript, e.g. v_i, is used to denote an element of the vector.

A vector may also be used to express a deficient matrix, in which case it is necessary to distinguish between a row vector and a column vector.