Theorem 10.4 The angle between vectors u and v in R” is

(a) acute, if u v > 0,

(b) obtuse, if u Y < 0,

(c) right, if u Y = 0.

When this angle is a right angle, we say that u and v are orthogonal. So,

vectors u and v are orthogonal if and only if u v = uivl + + U.V. = 0, a

simple check indeed.

We have taken some liberties with the case where one of the vectors is zero.

When this occurs, 0 is not defined. However, we will run into no difficulties with

if we simply watch for zero vectors.

the concept of orthogonality

Finally, we use Theorem 10.3 to derive a basic property of length or norm-

the triangle inequality. This rule states that any side of a triangle is shorter than

the sum of the lengths of the other two sides. Intuitively, it follows from the fact

that the straight line segment gives the shortest path between any two points in

R”. In vector notation, we want to prove that

IIn + ˜11 5 IIn + llvll for all II, v in R”

Figure 10.22 illustrates the equivalence of this analytic formulation with the ahove

statement about triangles.

Figure

of a

n, v, and u + Y are ihree sides trim&.

10.22

[I 0.41 LENGTH AND INNER PRODUCT IN R” 219

Theorem 10.5 For any two vectors u, v in R”,

Ilo + 41 5 lbll + Ibll. (6)

=coso51

“˜Y

IMI llvll

™ by Theorem 10.3. Therefore,

llul12 + 2(u v) + llYl12 5 lld + 2llullllvll + Ilvll˜,

“. ” + ” ” + Y ” + ” ” 5 (Ilull + Ilvll)˜,

(0 + v) (u + v) 5 mdl + llvll)*,

IIU + VII2 5 (Ilull + llvll)2,

llu + VII 5 Ilull + IIVII. w

We will use the triangle inequality (6) over and over again in our study of

Euclidean spaces. Just about every mathematical statement involving an inequality

requires the triangle inequality in its proof. The next theorem presents a variant of

the triangle inequality which we will also use frequently in our analysis, especially

when we want to derive a lower bound for some expression. To understand this

result mwe fully, you should test it on pairs of real numbers, especially pairs with

oppose signs.

Theorem 10.6 For any two vectors x and y in R”,

I llxll - IIYII I 5 IIX - YII.

Proof Apply Theorem 10.5 with u = x - y and v = y in (6), to obtain the

wwality llxll s llx - yll + Ilyll, or

llxll - IIYII 5 Ilx YII. (7)

1™

Now apply Theorem 10.5 with u = y - x and v = x in (6) to obtain the

I. ˜.

UqJal˜tY IIYII 5 IIY - XII + IIXII, or

IIYII - llxll 5 IIY - XII = l/x YII- (8

EUXDEAN SPACES [lOI

220

Inequalities (7) and (8) imply that

I llxll - llyll I 5 IIX YII. w

The three basic properties of Euclidean length are:

(1) ((˜11 2 0 and Ilull = 0 only when” = 0.

(4 lhll = lrlllull

(3) IIU + VII 5 Ilull + Ilvll.

Any assignment of a real number to a vector that satisfies these three prop-

erties is called a norm. Exercise 10.16 lists other mxms that arise naturally in

applications. We will say more about norms in the last section of Chapter 29.

EXERCISES

ltl.10 Find the length of the following vectors. Draw the vectors for a through g:

E) (-I, -11,

b) (0, -3), c) (I, 1, 11, 4 (3,3),

a) (3, 4))

&!) (A a i) (3. 0, 9 0, 0).

h) (I, 2, 3, 4),

f) (1, Z3)>

10.11 Find the distance from P to Q, drawing the picture wherever possible:

Q(7. 7):

a) PO3 01, Q(3, -4); h) P(1, -I),

Q(L 2); d ) P(1, I, -l), Q(Z, -1.5):

c) w 3,

e) P(I, 2. 3, 4)s Q(l, 01, - 1, 0).

10.12 For each ofthc following pairsofvectars, first determine whether the angle between

them is acute, obtuse; or right and then calculate this angle:

v = (2 -8):

” = (2, 2);

0) ” = (1, m h) ” = (4, I),

d) u = (I, -LO), v = (1,2, 1):

” = (I, 2, 1):

c) ” = (1. 1. O),

e, ” = (1,0,0.0,0), ” = (1, I, I, I, 1).

10.13 For each of the following vectors, find n vector of length I which points in the same

direction. a) (3,4), h) (6, 0). c) (1, I, 1). d) (m I, 2, -3).

10.14 For each of the vectors in the last exercise, find a vector of length five which points

in the opposite direction.

10.15 Prove that IIu ˜11™ = Ilull™ 2”. v + 11˜11˜.

10.16 a) In view uf the last paragraph in this section, prove that each of the lollowing ii

a norm in R™:

lll(u1, u2)lll = IUII + Id

IllCut, u˜)lll = max{lu,I, lu2N.

h) What arc the analogous narms in R”˜!

10.17 Provide a complete and careful proof of Theorem 10.2.

221

[lo.41 LENGTH AND INNER PRODUCT IN R”