54 CHAPTER 5. FUNDAMENTALS
2
6 • (2,6)
−8
3•(−8,3)
Notice how you can identify a point shown in the plane with the ordered pair (2,6) .You go to the right a distance of 2 and then up a distance of 6. Similarly, you can identifyanother point in the plane with the ordered pair (−8,3) . Go to the left a distance of 8 andthen up a distance of 3. The reason you go to the left is that there is a − sign on the eight.From this reasoning, every ordered pair determines a unique point in the plane. Conversely,taking a point in the plane, you could draw two lines through the point, one vertical and theother horizontal and determine unique points x1 on the horizontal line in the above pictureand x2 on the vertical line in the above picture, such that the point of interest is identifiedwith the ordered pair (x1,x2) . In short, points in the plane can be identified with orderedpairs similar to the way that points on the real line are identified with real numbers. Nowsuppose n = 3. As just explained, the first two coordinates determine a point in a plane.Letting the third component determine how far up or down you go, depending on whetherthis number is positive or negative, this determines a point in space. Thus, (1,4,−5) wouldmean to determine the point in the plane that goes with (1,4) and then to go below thisplane a distance of 5 to obtain a unique point in space. You see that the ordered triplescorrespond to points in space just as the ordered pairs correspond to points in a plane andsingle real numbers correspond to points on a line.
You can’t stop here and say that you are only interested in n ≤ 3. What if you wereinterested in the motion of two objects? You would need three coordinates to describewhere the first object is and you would need another three coordinates to describe wherethe other object is located. Therefore, you would need to be considering R6. If the twoobjects moved around, you would need a time coordinate as well. As another example,consider a hot object which is cooling and suppose you want the temperature of this object.How many coordinates would be needed? You would need one for the temperature, threefor the position of the point in the object and one more for the time. Thus you would needto be considering R5. Many other examples can be given. Sometimes n is very large. Thisis often the case in applications to business when they are trying to maximize profit subjectto constraints. It also occurs in numerical analysis when people try to solve hard problemson a computer.
There are other ways to identify points in space with three numbers but the one pre-sented is the most basic. In this case, the coordinates are known as Cartesian coordinatesafter Descartes1 who invented this idea in the first half of the seventeenth century. I willoften not bother to draw a distinction between the point in n dimensional space and itsCartesian coordinates.
1ReneĢ Descartes 1596-1650 is often credited with inventing analytic geometry although it seems the ideas wereactually known much earlier. He was interested in many different subjects, physiology, chemistry, and physicsbeing some of them. He also wrote a large book in which he tried to explain the book of Genesis scientifically.Descartes ended up dying in Sweden.