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The Real Line

The simplest geometric object is the number line-the geometric realization of

the set of all real numbers. The number line was defined carefully at the beginning

of Chapter 2. Every real number is represented by exactly one point on the line.

and each point on the line represents one and only one numbcr. Figure 10.1 shows

вЂњart of a number line.

Figure

10.1

The Plane

In some of our economic examples. we have used pairs of numbers to repre-

sent economic objects, for example, consumption bundles in Chapter 1. Pairs

of numbers also have a geometric representation, called the Cartesian plane or

Euclidean Z-space, and written as R2. To depict RвЂ™, first draw two pcrpcndicular

number lines: one horizontal to represent the first component x1 of the pair

(x1, x2) and the other vertical to represent the second component x2 of (x,, x2). The

unit length is usually the same along each line (although it need not be). These two

200 EVCLIDEAN SPACES I1 01

Figure

10.2

number cdlled They

lines arc coordinate axes. intersect at their origins. Figure

10.2 shows how each point in the plane is identified with a unique pair of numbers.

We have used the Cartesian plane in Chapter 2 to draw graphs of functions of one

variable.

A point p in the Cartcsiun plane represents a pair of numbers (u. h) as follows:

draw a vertical line eI and a horizontal line e2 through the point p. The vertical

line CTOSXS the x,-axis at u. and the horirontal lint crosses the x-axis at h. WC

associate the pair (a, h) with the point p. To go the ˜thcr way ˜ to End the point

p which represents the pair (a, l)-find a on the I,-axis. and through it draw

the vertical line <, Find b on the I,-axis. and through it draw the horizontal line

P:. The intersection of the two lines XI and fZ2 is the point p. which WC will

sometimes write p(u. h). The number a is called the x,-coordinate of p. and h

as

is called the of In Figure 10.3 we show a numhcr of points and

xl-coordinate p.

their coordinates.

Figure

10.3

110.1 I POINTS AND вЂњECTOKS IN EVCLlDEAN SPACE 201

The point of intersection of the horizontal and vertical number lines is our

reference point for measuring the location of p. It is called the origin, and we

denote it by the symbol 0, since it is represented by the pair (0, 0).

Three Dimensions and More

Similarly, one can visualize 3-dimensional Euclidean space RвЂќ by drawing three

mutually perpendicular number lines. As before, each one of these number lines

is called a coordinate axis: the xl-axis, the x2-axis, and the x,-axis, respectively.

One usually draws the x2-axis as the horizontal axis and the .x-axis as the vertical

axis on the plane of the page and then pictures the x,-axis as coming out of the

page toward one, as in Figure 10.4.

The process of identifying a point with a particular triple of numbers uses

the techniques that we used in RвЂ™. The process is illustrated in Figure 10.5. To

/вЂ™

/вЂ™

/вЂ™

/вЂ™

Figure

10.4

Figure

Thp pod p with coordinates (a, b, c). 10.5

202 EUCLIDEAN SPAACES [I 01

find the point represented by the triple (a, b, c). forget about a for a moment, and

locate the point representing (b, c) in the xzx3-plane - the plane of the page. This

is a Z-space exercise that we already know how to do. From the point (4 c) in

the plane of the page, move a units in the direction parellel to the xl-axis. March

out of the page if a is positive, and march behind the page if a is negative. If a

is 0, remain where you are. The point p at which you finish represents (a, b, c)

and is sometimes denoted by p(a, b, c). We could have just as easily started in the

x,.x-plane and then moved b units to the right (for positive b), or in thex,xz-plane

and then moved c units up (for positive c). Check to see that you end up at the

same point no matter which method you use.

Finding the coordinates that describe a particular point p is just as easy.

Starting from p, move parallel to the x,-axis until you reach x@-plane. The

distance moved is a; it is positive if the move was into the page and negative if

the move was out toward you. The coordinates b and c are now found using the

Z-space technique. Again, the answer is independent of which plane you head for

first. This description and the accompanying diagram (Figure 10.5) is an example

of a situation where a picture is worth a thousand words.

Of course, we cannot draw geometric pictures of higher-dimensional Euclidean

spaces, but we can use our pictures of RвЂ™, R2, and R3 to guide our intuition. We

will see that the formulas describing geometric objects and their properties in

RвЂќ and R-вЂ™ generalize readily to higher dimensions. The real lint RвЂ™ consists of

single numbers. The plane R2 consists of ordered pairs of numbers. We say

ordered pairs because the order of the numbers matters; (I, 0) is not the same as

(0, 1). Euclidean n-space consists of ordered n-tuples of numbers- ordered lists

of n numbers. For example, Euclidean &pace contains ordered triples (a, b, c) of

numbers. Euclideen S-space contains ordered S-tuples (u. b, c, d, e). Euclidean n-

space is usually referred to as RвЂќ. The number n in RвЂќ refers to how many numbers

are needed to describe each location. It is called the dimension of RвЂќ. Thus RS

has 5 dimensions, while RвЂ™ has only tvw dimensions. Each space will have its

origin, the point with respect to which we make our coordinate measurements. As

we did in R2. we will always refer to the origin hy the symbol 0.

10.2 VECTORS

Euclidean spaces are useful for modeling a wide variety of economic phenomena

because n-tuples of numbers have many useful interpretations. Thus far we have

emphasized their interpretation as locations, or points in n-space. For example, the

point (X,2) represents a particular location in the plane, found by going 3 units to

the right and 2 units up from the origin. This is just the way we use coordinates on

a map of a country to find the location of a particular city. We USC coordinates to

describe locations in exactly the same way in higher dimensions. Many economic

applications require us to think of n-tuples of numhcrs as locations. For example.

we think of consumption hundles as locations in commodity spuce.

We can also interpret n-tuplrs as displacements. This is a useful way of

thinking about vectors for doing calculus. We picture these displacements as

r10.21 вЂњECTORS 2 0 3

arrows in RвЂќ. The displacement (3,2) means: move 3 units to the right and 2 units

up from your current location. The tail of the arrow marks the initial location; the

head marks the location after the displacement is made. In Figure 10.6, each arrow

represents the displacement (3,2), but in each case the displacement is applied to

a different initial location.

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