Principles of Mathematics in Operations Research

34 3 Orthogonality

Furthermore, we know the following relations:

y/n

< x <

loo —

12'

l 2 < li<IMI^2 -

y/n

< X I < X 1 •

Remark 3.1.3 The good-old Euclidian distance is the l<z norm that indicates

the bird-flight distance. In Figure 3.1, for instance, a plane's trajectory between

two points (given latitude and longitude pairs) projected on earth (assuming

that it is flat!) is calculated by using the Pythagoras Formula. The rectilinear

distance (l\ norm) is also known as the Manhattan distance. It indicates the

mere sum of the distances along the canonical unit vectors. It assumes the

dependence of the movements along with the coordinate axes. In Figure 3.1,

the length of the pathway restricted by blocks, of the car from the entrance of a

district to the current location is calculated by adding the horizontal movement

to the vertical. The Tchebychev's distance (1^) simply picks the maximum

distance among all movements along the coordinate axes, and thus, assumes

total independence. The forklift in Figure 3.1 can move sideways by its main

engine, and it can independently raise or lower its fork by another motor. The

total time it takes for the forklift to pick up an object 10m. away from a rack

lying on the floor and place the object on a rack shelf 3m. above the floor is

simply the maximum of the travel time and the raising time. A detailed formal

discussion of metric spaces is located in Section 10.1.

I

Jl

-x1 •

Fig. 3.1. Metric examples: ||.| 2 ' 11*111 ' ll-lloo

Definition 3.1.4 The length \x\ 2 of a vector x in K™ is the positive square
root of