Simple Nature - Light and Matter

3.4.3 VectorsRemember the title of this book? It would have been possible toobtain the result of example 55 by applying the Pythagorean theoremto dx, dy, and dz, and then dividing by dt, but the rotationalinvariance approach is simpler, and is useful in a much broadercontext. Even with a quantity you presently know nothing about,say the magnetic field, you can infer that if the components of themagnetic field are B x , B y , and B z , then the physically useful wayto talk about the strength of the magnetic field is to define it as√B 2 x + B 2 y + B 2 z. Nature knows your brain cells are precious, anddoesn’t want you to have to waste them by memorizing mathematicalrules that are different for magnetic fields than for velocities.When mathematicians see that the same set of techniques isuseful in many different contexts, that’s when they start makingdefinitions that allow them to stop reinventing the wheel. The ancientGreeks, for example, had no general concept of fractions. Theycouldn’t say that a circle’s radius divided by its diameter was equalto the number 1/2. They had to say that the radius and the diameterwere in the ratio of one to two. With this limited number concept,they couldn’t have said that water was dripping out of a tank ata rate of 3/4 of a barrel per day; instead, they would have had tosay that over four days, three barrels worth of water would be lost.Once enough of these situations came up, some clever mathematicianfinally realized that it would make sense to define somethingcalled a fraction, and that one could think of these fraction thingiesas numbers that lay in the gaps between the traditionally recognizednumbers like zero, one, and two. Later generations of mathematiciansintroduced further subversive generalizations of the numberconcepts, inventing mathematical creatures like negative numbers,and the square root of two, which can’t be expressed as a fraction.In this spirit, we define a vector as any quantity that has bothan amount and a direction in space. In contradistinction, a scalarhas an amount, but no direction. Time and temperature are scalars.Velocity, acceleration, momentum, and force are vectors. In one dimension,there are only two possible directions, and we can use positiveand negative numbers to indicate the two directions. In morethan one dimension, there are infinitely many possible directions, sowe can’t use the two symbols + and − to indicate the direction ofa vector. Instead, we can specify the three components of the vector,each of which can be either negative or positive. We representvector quantities in handwriting by writing an arrow above them,so for example the momentum vector looks like this, ⃗p, but the arrowlooks ugly in print, so in books vectors are usually shown inbold-face type: p. A straightforward way of thinking about vectorsis that a vector equation really represents three different equations.For instance, conservation of momentum could be written in termsSection 3.4 Motion In Three Dimensions 193