стр. 34 |

Q

"

//

P

Figure

The displacement (3,2). 10.6

For example, the tail of the displacement labeled v in Figure 10.6 is at the

location (3, I), and the head is at (6, 3). We will sometimes write @ for the

displacement whose tail is at the point P and head at the point Q. Two arrows

represent the same displacement if they are parallel and have the same length and

direction. For our purposes, two such arrows are equivalent; regardless of their

different initial and terminal locations, they both represent the same displacement.

The essential ingredients of a displacement are its magnitude and direction.

How do we assign an n-tuple to a particular arrow? We measure how far we

have to move in each direction to get from the tail to the head of the arrow. For

example, consider the arrow v in Figure 10.6. To get from the tail to the head

we have to move 3 units in the x,-direction and 2 units in the x2-direction. Thus

v must represent the displacement (3,2). More formally, if a displacement goes

from the initial location (a, h) tu the terminal location (c, d), then the move in the

xl-direction is c - a, since a + (c a) = c; and the move in the xz-direction is

d-h, since h + (d b) = d. Thus the displacement is (c a, d - b). This method

of subtracting corresponding coordinates applies to higher dimensions as well.

The displacement from the point ˜(a,, a˜, , a.) to the point q(bl, bz,. ., b,) in

RвЂќ is written

rq = (b, - a,, hi a˜, ., b, a,)

Figure 10.6 illustrates that there are many (3,2) displacements. In any given

discussion, all the displacements will usually have the same initial location (tail).

Often, this initial location will naturally be 0, the origin. From this initial location,

Figure

10.7 Some displacements in the plane.

the displacement (3,2) takes us to the location (3,Z). With this вЂњcanonical repre-

sentationвЂќ of displacements, we can think of locations as displacements from the

origin. Several different displacements are shown in Figure 10.7.

We have just seen that the very different concepts of location and displace-

ment have a common mathematical representation as n-tuples of numbers. These

concepts act alike mathematically, and so we give them a common name: vectors.

Some books distinguish between locations and displacements by writing a

location as a row vector (a. h) and a displacement as a column vector i This

0

approach is unwieldy and unnecessary. From now on we will use the word вЂњvectorвЂќ

to refer to both locations and displacements. It will either be explicitly mentioned,

or clear from the context, whether locations OT displacements are meant in any

particular discussion.

EXERCISES

10.1 Draw a number line and locate (approximately) the points I, 31вЂ™2. 2. 45. 71. and

-n/2.

10.2 Draw a Cartesian plane and locate on it the following points: (I, 1j. (˜ I /2.3/2).

(O,O), (0. -4); cr. 4).

10.3 Draw a plane, and show the path you would traverse were you to start at (˜ I, 31,

displace yourself first by the vector (I, -31, and then hy the vector C-1, -3).

For the points P and Q listed below, draw the corresponding displacement vector

10.4

?$ and compute the corresponding n-tuple for zsi:

a n d Qcl, I).

a n d Q(2. -I).

0) P(O.0) h) P0.2)

a n d Q(3. I).

c) P(3.2) and Q(5.3). 4 P(O. I)

f, P(O.l,Ol a n d Q(2,-1.3).

e) P(O.O.0) a n d Q(l.2.41,

205

I1 0.31 THE ALGEBRA OF VECTORS

10.3 THE ALGEBRA OF VECTORS

There are four basic algebraic operations for the real numbers, RвЂ™: addition,

subtraction, multiplication and division. This section introduces the three basic

algebraic operations on higher-dimensional Euclidean spaces: vector addition and

subtraction and scalar multiplication.

Addition and Subtraction

We add two vectors just as we add two numbers. We simply add separately the

corresponding coordinates of the two vectors. Thus

(3,2) + (4, 1) = (7,3),

(Xl> x2, Xi) + h Y2, Y3) = (XI + YI> x2 + YZ, 13 + Y?).

Notice that we can only add together two vectors from the same vector space.

The sum (2, 1) + (3,4, I) is not defined, since the first vector lives in Rz while

the second vector lives in R3. Furthermore, the sum of two vectors from RвЂќ is a

vector, and it lives in RвЂќ. When we add (3,5, I, 0) + (990, 1) from R4, we get

the vector (3, 5, 1, I) which is also in R4.

To develop a geometric intuition for vector addition, it is most natural to think

of vectors as displacement arrows. If u = (a, 6) and v = (c, d) in R*, then we

want u + v to represent a displacement of a + c units to the right and h + d

units up. Intuitively, we can think of this displacement as follows: Start at some

initial location. Apply displacement u. Now apply displacement v to the terminal

location of the displacement u. In other words, move v until its tail is at the head

of u. Then, u + v is the displacement from the tail of u to the head of v, as in

Figure 10.8. Verify that u + Y, as drawn, has coordinates (u + L:, h + d).

Figw

10.8

206 EUCUDEAN [I 01

SPACES

I

Figure

10.9 u+v=v+u.

Figure 10.9 shows that it makes no difference whether we think of u + v as

displacing first by u and then by Y OI first by v and then by II. Since the hvo arrows

representing u in Figure 10.9 are parallel and have the same length and similarly

for the two representations of v, the quadrilateral in Figure 10.9 is a parallelogram.

Its diagonal represents both u + v and v + u. Formally, Figure 10.9 shows that

u + v = v + u; vector addition, like addition of real numbers, is commutative.

One can use the parallelogram in Figure 10.9 to draw u + Y while keeping the

tails of u and v at the same point. First, draw the complete parallelogram which

has u and Y as adjacent sides, as in Figure 10.9. Then, take u + v as the diagonal

of this parallelogram with its tail at the common tail of u and v. Physicists use

displacements vectors to represent forces acting at a given point. If vectors u and

v represent two forces at point P, then the vector u + v represents the force which

results when both forces are applied at P at the same time.

Vector addition obeys the other rules which the addition of real numbers obeys.

These are: the associative rule, the existence of a zem (an additive identity), and

the existence of an additive inverse. The zero vector is the vector which represents

no displacement at all. Analytically we write

0 = (0, 0, , 0).

Geometrically, it is a displacement ?? having the sane terminal point as initial

point. Check both algebraically and geometrically that u + 0 = u.

If u = (a,, a?, , an), then the negative of u, written --u and called вЂњminus uвЂќ,

is the vector (pa,, ˜a˜, , -an). Geometrically, one interchanges the head and

стр. 34 |