Einstein 1938: Vectors

Reference: Evolution of Physics

This paper presents Chapter I, section 3 from the book THE EVOLUTION OF PHYSICS by A. EINSTEIN and L. INFELD. The contents are from the original publication of this book by Simon and Schuster, New York (1942).

The paragraphs of the original material (in black) are accompanied by brief comments (in color) based on the present understanding.  Feedback on these comments is appreciated.

The heading below is linked to the original materials.



All motions we have been considering are rectilinear, that is, along a straight line. Now we must go one step farther. We gain an understanding of the laws of nature by analyzing the simplest cases and by leaving out of our first attempts all intricate complications. A straight line is simpler than a curve. It is, however, impossible to be satisfied with an understanding of rectilinear motion alone. The motions of the moon, the earth and the planets, just those to which the principles of mechanics have been applied with such brilliant success, are motions along curved paths. Passing from rectilinear motion to motion along a curved path brings new difficulties. We must have the courage to overcome them if we wish to understand the principles of classical mechanics which gave us the first clues and so formed the starting-point for the development of science.

Rectilinear motion is a special case of curvilinear motion.

Let us consider another idealized experiment, in which a perfect sphere rolls uniformly on a smooth table. We know that if the sphere is given a push, that is, if an external force is applied, the velocity will be changed. Now suppose that the direction of the blow is not, as in the example of the cart, in the line of motion, but in a quite different direction, say, perpendicular to that line. What happens to the sphere? Three stages of the motion can be distinguished: the initial motion, the action of the force, and the final motion after the force has ceased to act. According to the law of inertia, the velocities before and after the action of the force are both perfectly uniform. But there is a difference between the uniform motion before and after the action of the force: the direction is changed. The initial path of the sphere and the direction of the force are perpendicular to each other. The final motion will be along neither of these two lines, but somewhere between them, nearer the direction of the force if the blow is a hard one and the initial velocity small, nearer the original line of motion if the blow is gentle and the initial velocity great. Our new conclusion, based on the law of inertia, is: in general the action of an external force changes not only the speed but also the direction of the motion. An understanding of this fact prepares us for the generalization introduced into physics by the concept of vectors.

In general the action of an external force changes not only the speed but also the direction of the motion.

We can continue to use our straightforward method of reasoning. The starting-point is again Galileo’s law of inertia. We are still far from exhausting the consequences of this valuable clue to the puzzle of motion.

Let us consider two spheres moving in different directions on a smooth table. So as to have a definite picture, we may assume the two directions perpendicular to each other. Since there are no external forces acting, the motions are perfectly uniform. Suppose, further, that the speeds are equal, that is, both cover the same distance in the same interval of time. But is it correct to say that the two spheres have the same velocity? The answer can be yes or no! If the speedometers of two cars both show forty miles per hour, it is usual to say that they have the same speed or velocity, no matter in which direction they are travelling. But science must create its own language, its own concepts, for its own use. Scientific concepts often begin with those used in ordinary language for the affairs, of everyday life, but they develop quite differently. They are transformed and lose the ambiguity associated with them in ordinary language, gaining in rigorousness so that they may be applied to scientific thought.

From the physicist’s point of view it is advantageous to say that the velocities of the two spheres moving in different directions are different. Although purely a matter of convention, it is more convenient to say that four cars travelling away from the same traffic roundabout on different roads do not have the same velocity even though the speeds, as registered on the speedometers, are all forty miles per hour. This differentiation between speed and velocity illustrates how physics, starting with a concept used in everyday life, changes it in a way which proves fruitful in the further development of science.

Science introduces the word velocity to account for both speed and the direction of motion.

If a length is measured, the result is expressed as a number of units. The length of a stick may be 3 ft. 7 in.; the weight of some object 2 Ib. 3 oz.; a measured time interval so many minutes or seconds. In each of these cases the result of the measurement is expressed by a number. A number alone is, however, insufficient for describing some physical concepts. The recognition of this fact marked a distinct advance in scientific investigation. A direction as well as a number is essential for the characterization of a velocity, for example. Such a quantity, possessing both magnitude and direction, is called a vector. A suitable symbol for it is an arrow. Velocity may be represented by an arrow or, briefly speaking, by a vector whose length in some chosen scale of units is a measure of the speed, and whose direction is that of the motion.

Mathematics introduces the word vector for a quantity possessing both magnitude and direction.

If four cars diverge with equal speed from a traffic roundabout, their velocities can be represented by four vectors of the same length, as seen from our last drawing. In the scale used, one inch stands for 40 mph. In this way any velocity may be denoted by a vector, and conversely, if the scale is known, one may ascertain the velocity from such a vector diagram.

If two cars pass each other on the highway and their speedometers both show 40 mph, we characterize their velocities by two different vectors with arrows pointing in opposite directions. So also the arrows indicating “uptown” and “downtown” subway trains in New York must point in opposite directions. But all trains moving uptown at different stations or on different avenues with the same speed have the same velocity, which may be represented by a single vector. There is nothing about a vector to indicate which stations the train passes or on which of the many parallel tracks it is running. In other words, according to the accepted convention, all such vectors, as drawn below, may be regarded as equal; they lie along the same or parallel lines, are of equal length, and finally, have arrows pointing in the same direction.

The next figure shows vectors all different, because they differ either in length or direction, or both. The same four vectors may be drawn in another way, in which they all diverge from a common point. Since the starting-point does not matter, these vectors can represent the velocities of four cars moving away from the same traffic roundabout, or the velocities of four cars in different parts of the country travelling with the indicated speeds in the indicated directions.

The consideration of vector is geometrical.

This vector representation may now be used to describe the facts previously discussed concerning rectilinear motion. We talked of a cart, moving uniformly in a straight line and receiving a push in the direction of its motion which increases its velocity. Graphically this may be represented by two vectors, a shorter one denoting the velocity before the push and a longer one in the same direction denoting the velocity after the push. The meaning of the dotted vector is clear; it represents the change in velocity for which, as we know, the push is responsible. For the case where the force is directed against the motion, where the motion is slowed down, the diagram is somewhat different. Again the dotted vector corresponds to a change in velocity, but in this case its direction is different. It is clear that not only velocities themselves but also their changes are vectors. But every change in velocity is due to the action of an external force; thus the force must also be represented by a vector. In order to characterize a force it is not sufficient to state how hard we push the cart; we must also say in which direction we push. The force, like the velocity or its change, must be represented by a vector and not by a number alone. Therefore: the external force is also a vector, and must have the same direction as the change in velocity. In the two drawings the dotted vectors show the direction of the force as truly as they indicate the change in velocity.

Not only the velocity, but the change in velocity, and the force responsible for that change, are also vectors.

Here the sceptic may remark that he sees no advantage in the introduction of vectors. All that has been accomplished is the translation of previously recognized facts into an unfamiliar and complicated language. At this stage it would indeed be difficult to convince him that he is wrong. For the moment he is, in fact, right. But we shall see that just this strange language leads to an important generalization in which vectors appear to be essential.

The concept of vector is essential for further development of this subject.


From a universal viewpoint motion is always curvilinear, but from a local viewpoint it appears as rectilinear. All particles in space are revolving around some axes. The smaller is the mass the larger is the radius of revolution per the cosmic geometry described at Motion & Force. The path of any particle is curvilinear, but as the radius increases the path appears more rectilinear.

An external force may change the particle’s speed and direction, but its mass will try to restore it back to its original speed and direction.

LAW OF INERTIA: Inertia is the internal force that resides in mass. This internal force is continuous, whereas, the external forces are intermittent. Therefore, this internal force prevails in the long run. Inertia not only keeps the motion uniform but it also brings the uniform velocity in line with the mass.


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