Not only Newton’s laws, but also the other laws of physics, so far as we know today, have the two properties which we call invariance (or symmetry) under translation of axes and rotation of axes. These properties are so important that a mathematical technique has been developed to take advantage of them in writing and using physical laws.

The foregoing analysis involved considerable tedious mathematical work. To reduce the details to a minimum in the analysis of such questions, a very powerful mathematical machinery has been devised. This system, called vector analysis, supplies the title of this chapter; strictly speaking, however, this is a chapter on the symmetry of physical laws. By the methods of the preceding analysis, we were able to do everything required for obtaining the results that we sought, but in practice we should like to do things more easily and rapidly, so we employ the vector technique.

We began by noting some characteristics of two kinds of quantities that are important in physics. (Actually, there are more than two, but let us start out with two.) One of them, like the number of potatoes in a sack, we call an ordinary quantity, or an undirected quantity, or a scalar. Temperature is an example of such a quantity. Other quantities that are important in physics do have direction, for instance velocity: we have to keep track of which way a body is going, not just its speed. Momentum and force also have direction, as does displacement: when someone steps from one place to another in space, we can keep track of how far he went, but if we wish also to know where he went, we have to specify a direction.

All quantities that have a direction, like a step-in space, are called vectors.

A vector is three numbers. In order to represent a step-in space, say from the origin to some particular point P whose location is (x,y,z), we really need three numbers, but we are going to invent a single mathematical symbol, r, which is unlike any other mathematical symbols we have so far used.¹ It is not a single number, it represents three numbers: x, y, and z. It means three numbers, but not really only those three numbers, because if we were to use a different coordinate system, the three numbers would be changed to x′, y′, and z′. However, we want to keep our mathematics simple and so we are going to use the same mark to represent the three numbers (x,y,z) and the three numbers (x′,y′,z′). That is, we use the same mark to represent the first set of three numbers for one coordinate system, but the second set of three numbers if we are using the other coordinate system. This has the advantage that when we change the coordinate system, we do not have to change the letters of our equations. If we write an equation in terms of x,y,z, and then use another system, we have to change to x′,y′,z′, but we shall just write r, with the convention that it represents (x,y,z) if we use one set of axes, or (x′,y′,z′) if we use another set of axes, and so on. The three numbers which describe the quantity in a given coordinate system are called the components of the vector in the direction of the coordinate axes of that system. That is, we use the same symbol for the three letters that correspond to the same object, as seen from different axes. The very fact that we can say “the same object” implies a physical intuition about the reality of a step-in space, that is independent of the components in terms of which we measure it. So, the symbol r will represent the same thing no matter how we turn the axes.

Now suppose there is another directed physical quantity, any other quantity, which also has three numbers associated with it, like force, and these three numbers change to three other numbers by a certain mathematical rule, if we change the axes. It must be the same rule that changes (x,y,z) into (x′,y′,z′). In other words, any physical quantity associated with three numbers which transform as do the components of a step-in space is a vector. An equation like

F = r

would thus be true in any coordinate system if it were true in one. This equation, of course, stands for the three equations

F_x = x, F_y = y, F_z = z,

or, alternatively, for

F_x′ = x′, F_y′ = y′, F_z′ = z′.

The fact that a physical relationship can be expressed as a vector equation assures us the relationship is unchanged by a mere rotation of the coordinate system. That is the reason why vectors are so useful in physics.

Now let us examine some of the properties of vectors. As examples of vectors, we may mention velocity, momentum, force, and acceleration. For many purposes it is convenient to represent a vector quantity by an arrow that indicates the direction in which it is acting. Why can we represent force, say, by an arrow? Because it has the same mathematical transformation properties as a “step in space.” We thus represent it in a diagram as if it were a step, using a scale such that one unit of force, or one newton, corresponds to a certain convenient length. Once we have done this, all forces can be represented as lengths, because an equation like

F = kr,

where k is some constant, is a perfectly legitimate equation. Thus, we can always represent forces by lines, which is very convenient, because once we have drawn the line, we no longer need the axes. Of course, we can quickly calculate the three components as they change upon turning the axes, because that is just a geometric problem.

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1- In type, vectors are represented by boldface; in handwritten form an arrow is used: r.