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Figure

10.15

(10.41 LENGTH AND INNER PKODVCTIN 211

RвЂќ

Next, consider the case where P and Q have the same n, -component. Say P is

(a, bl) and Q is (a, b), as in Figure 10.14. Here, the distance is naturally 19 - b, 1.

Finally, we consider the general case, as pictured in Figure 10.15. To compute

the length of line 0 joining points P(a,, bl) and Q(az, bz), mark the intermediate

point R(aZ, bl). Let m be the (horizontal) line segment from P(a,, bl) to R(a2, b,)

and let n be the (vertical) line segment from Q(a, b,) to R(q, b,). The cor˜e-

sponding triangle PRQ is a right triangle whose hypotenuse is the line segment X.

Apply the Pythagorean Theorem to deduce the length of 4:

(length-e)вЂ™ = (length m)вЂ™ + (length n)вЂ™

= la, a# + lb, - b#.

Taking the square root of both sides of this equation gives

Ilmll = length4 = &a, - a# + (b, - b#. (1)

We can apply this argument to higher dimensions, as pictured in Figure 10.16.

To find the distance from P(a,, bl, cl) to Q(az, bz, cz) in R3, we use the point

R(q, b2, c,), which has the same .x-coordinate as P and the same x1- and x2-

coordinates as Q. Since P and R have the same .x-coordinate, the segment PR lies

on the x3 = c, plane, which is parallel to the x,x2-plane (x3 = 0). Since Q and R

have the same xl- and x+oordinates, segment QR is parallel to the q-axis and

therefore perpendicular to the segment PR. Therefore, APRQ is a right triangle

with hypotenuse PQ. By the Pythagorean Theorem,

IIPQIIвЂ™ = IlPRf + llRQl12. (4

_; 5pc$7J

R

I

Fig

Computing the length of line PQ in R3, 10.˜

212 EUCLIDEAN SPACES 1101

Since RQ is parallel to the ,x-axis, its length is simply 1˜2 - c, I. To find the length

of PR, we work in the two-dimensional plane through PR parallel to the x1x2-

plane. Note that if S = (a*, bl, q), PS is parallel tc˜ the xl-axis and therefore has

length Ia2 ˜ a, I, and SR is parallel to the x2-x& with length lb2 ˜ bl 1. Applying

the Pythagorean Theorem to right triangle PSR yields:

llPRl12 = lIPSlIz + llSRll*

= Ia2 - al I* + lb2 - b, I*.

Substituting this into Equation (2) yields:

llPQl12 = laz - allвЂ™ + lb2 - b, I2 + Icz - c, 1вЂ™

Therefore, the distance from P to Q is

IlPQll = (a˜ ˜ al)вЂ™ + (bz ˜ b# + (CL - c#. (3)

Formulas (1) and (3) generalize readily to points in higher dimensional Eu-

clidean spaces. If (x1, .x2,. , x,J and (y,, yz, _, y,J are the coordinates of x and

y. respectively, in Euclidean n-space, then the distance between x and y is

&I - YI Y + (x2 - YZY + + (4, ˜ Y.Y

We will use this same formula whether we think of x and y as points or as

displacement vectors. Recall that x y is the vector joining points x and y and its

length IIx y/I is the same as the distance between these two points. Thus, it is

natural to write

In particular. if we take y to be 0, then the distance from the point x = (x,, , I,,)

to the origin or the length of the vector x is

We can now make more prccisc the effect of scalar multiplication on the length

of a vector Y. If I is a positive scalar, the length of IV is r times the length of v.

If I is a negative scalar, the length of IV is 111 times the length of v. This can be

summarized as follows.

1Theorem 10.1 /IrvII = 111 llvll for all I in RвЂ™ and v in RвЂќ. I

ilO. LENGTH AND INNER PRODUCT IN RвЂќ 213

Proof

I

Ilrh ., hz)II = Il(rv,, rvz, ., Yz)II

= (rv,)2 + + (rv,,)˜

=JW

n

= IrIJm,s i n c e fi = lrl.

Given a non-zero displacement vector v, we will occasionally need to find a

vector w which points in the same direction as v, but has length 1. Such a vector w

is called the unit vector in the direction of v, or sometimes simply the direction

of v. To achieve such a vector w, simply premultiply v by the scalar I = h,

because

Example 10.1 For example, the length of (1, -2, 3) in RвЂќ is

I

ll(l, -2,3)ll

I = 412 + (-2): + 32 = &i.

I I\ вЂ˜:d vector which points in the ˜ilmc direction as (I, -2, 3) but has length 1.

The Inner Product

We have learned how tc˜ add and subtracl two veckxs and how to compute the

distance between than. In this section we introduce another operation on pairs of

vectors, the Euclidcan inner product. This operation assigns a number to each pair

of vectors. We will see that it is connected 10 the notion of вЂњangle between two

vectors,вЂќ and therefore is useful for discussing geometric problems.

Definition Let u = (u,, u,,) and v = (v,, , v,,) be two vectors in RвЂќ. The

Euclidean inner product of u and v, written as u v, is the numhrr

214 EUCLIDEAN SPACES I101

Because of the dot in the notation, the Euclidean inner product is often called

the dot product. To emphasize that the result of the operation is a scalar, the

Euclidean inner product is also called the scalar product. In the exercises to this

section, we introduce the oвЂќter product or cross product as a way of multiplying

two vectors in R3 to obtain another vector in R3.

Example 10.2 IfвЂќ = (4, ˜1,2) and v = (6,3, ˜4), then

вЂњ.v=4.6+(-1).3+2.(˜4)= 13.

˜

The following theorem summarizes the basic analytical properties of the inner

product-properties that we will вЂњse often in this text. Its proof is straightforward

and is left as an exercise. Work oвЂќt the relationships in this theorem to build up a

working knowledge of inner product.

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