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the algebraic and geometric points of view are consistent and that u + (mu) = 0.

In the real numbers, subtraction is defined by the equation a - b = a + (-b).

We can use the same rule to define subtraction for vectors. Thus

(4,3,5) - (1,3,2) = (4, $5) + (PI, -3, -2)

= (4 - 1,3 3,s 2)

= (3, 0, 3).

207

DO.31 THE ALGEBRA OF VECTORS

More generally, for vectors in RвЂќ,

( 01, al>. a 4 - (bl, bz, , b,) = (al - b,, az ˜ bz,. >a. ˜ b,).

Geometrically we think of subtraction as completing the triangle in Figure 10.8.

Given u and u + v, find v to make the diagram work. Put another way, x - y is

that vector which, when added to y, gives x. Subtraction finds the missing leg of

the triangle in Figure 10.11˜.

Figure

10.10

of x ˜ y

Geometric representation

Scalar Multiplication

It is generally not possible to multiply two vectors in a nice way so as to generalize

the multiplication of real numbers. For example, coordinatewise multiplication

does not satisfy the basic properties that the multiplication of real numbers satisfies.

For one thing, the coordinatewise product of two nonzero vectors, such as (1,0) and

(0, 1). could be the zero vector. When this happens, division, the inverse operation

to multiplication. cannot bc detined. However, there is a vector space operation

which corresponds to statements like, вЂњgo twice as farвЂќ or вЂњyou are halfway

there.вЂќ This operation is called scalar multiplication. In it we multiply a vector,

cuordinatewise, by a real number. or scalar. If r is a scalar and x = (x,, , x˜) is

a vector, then their product is

r x = (TY,. ., rx,.x.

For example. 2 (I, I) = (2,2), and 4 (-4,2) = (-2, I).

Geometrically. scalar multiplication of a displacement vector x by a non-

negative scalar r corresponds to stretching or shrinking x hy the factor I without

changing its direction, as in Figure 10.11. Scalar multiplication by a negative

scalar causes not only a change in the length of a vector hut also a reverse in

direction.

In the algebra of the real numbers, addition and multiplication are linked by

the distributive laws:

a (b + c) = oh + ac (a + h) c = UC + bc.

and

EUCLIDEAN SPACES I1 01

208

-2x

Figure

10.11 Scalar multiplication in the plane.

There are distributive laws in Euclidean spaces as well. It is easy to see that

vector addition distributes over scalar multiplication and that scalar multiplication

distributes over vector addition:

(N) (r + .s)u = ru + su for all scalars r; s and vectors u.

(h) ,.(u + v) = IвЂќ + IV for all scalars I and vectors u, v

Any set of objects with a vector addition and scalar multiplication which

vector space. The

satisfies the rules we have outlined in this section is called a

elements of the set are called vectors. (The operations of vector addition and scalar

multiplication are the operations of matrix addition and scalar multiplication of

matrices, respectively. applied to I X n or II X I matrices. as defined in Section

I of Chapter 8. The scalar product of the next section will alw correspond to a

matrix operation.)

L10.41 LtNGiH AND iNNER PRODUCT IN RвЂќ 209

e

вЂќ + ,вЂќ + w) Figure

10.12

вЂќ + (v + w) = (u + v) + w.

10.4 LENGTH AND INNER PRODUCT IN RвЂќ

Among the key geometric concepts that guide our analysis of two-dimensional

economic models are length, distance and angle. In this section, we describe the

n-dimensional analogues of these concepts which WC will use for more complex,

higher dimensional economic models.

When we build mathematical models of economic phenomena in Euclidean

spaces, we will often be interested in the geometric properties of these spaces,

for example, the distance between two points or the angle between two vectors.

In this section we develop the analytical tools needed to study these properties.

In fact. all the geometrical results of planar (that is. two-dimensional) Euclidean

geometry can he derived using purely analytical techniques. Furthermore, these

analytic techniques arc all we have for generalizing the results of plane geometry

to higher-dimensional Euclidcan spaces.

Length and Distance

The most basic geometric property is distance or length. IfP and Q are two points

in RвЂќ, WC write FL) for the line segment joining P tu Q and PQ for the vector from

P to Q.

The length of line segment PQ is denoted by the symbol IlPQll. The

Notation

vertical lines draw attention to the analogy of length in the plane with absolute

value in the lint.

We now dcvclop a formula for IIPQII. or equivalently, f<,r the distance between

points P and Q. First, consider the case where P and Q lit in the plane R2 and have

the same x+xxndinate. We have pictured this situation in Figure 10.13, where

P has coordinates (a,. /I) and Q has coordinates (al, h). The length of this line is

clearly the length of the line segment connecting (I, and a? on the .x-axis. Since

length is alway\ a positive number, the length of this segment on the x,-axis is

simply [(I˜ (I, 1. We conclude that IlPQll = Ia2 n, I. as in Figure 10.13.

EUCLIDEAN SPACES 1101

210

Figure

llrnll = la2

10.13

Figure

10.14

I .,il

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