in a plane; namely, if the position of a relative to A be denoted by the directed step or vector AG, it may also be expressed by the sum of any number of directed steps, the start of the first of such steps being at a and the finish of the last at G (see fig. 56). We may write this result symbolically:

AGAB + BC + CD + DE + EF + FG.

It will be at once obvious that in our example as to finding the 'George,' the stranger might have been directed by an entirely different set of instructions to

his goal. In fact, he might have been led to make extensive circuits in or about the town before he reached the place he was seeking. But, however he might get to G, the ultimate result of his wanderings would be what he might have accomplished by the directed step AG supposing no obstacles to have been in his way (or, as the crow flies'). Hence we see that with our extended conception of addition any two zigzags of directed steps, ABCDEFG and A B′ C'D' E' F'G (which may or may not contain the same number of component steps), both starting in A and finishing in o,

must be looked upon as equivalent instructions; or, we must take

AB BC + CD + DE + EF + FG = AG = A B' + B'C' + C'D' + D'E' + E'F' + F'G. In other words, two sets of directed steps must be held to have an equal sum, when, their starts being the same, the steps of both sets will, added vector-wise have the same finish.

Now let us suppose our stranger were unconsciously standing in front of the 'George' when he asked his question as to its whereabouts, and further let us suppose that the person who directed him gave him a perfectly correct instruction, but sent him by a properly chosen set of right and left turnings a considerable distance round the town before bringing him back to the point a from which he had set out. In this case we must suppose the George' not to be at the point G, but at the point A. The total result of the stranger's wanderings having brought him back to the place from which he started can be denoted by a zero step; or we must write (fig. 56)—

6

AB+ BC + CD+DE+ E F + F G + GA = 0

We may read this in words: The sum of vector steps which form the successive sides of a closed zigzag is zero. Now we have found above that

AB+ BC + CD + DE+EF + FG AG

(ii)

Hence, in order that these two statements (i) and (ii) may be consistent, we must have GA equal to AG, or

AGGA 0.

This is really no more than saying that if a step be taken from A to G, followed by another from G to A, the total operation will be a zero step. Yet the result is

interesting as showing that if we consider a step from A to G as positive, a step from G to A must be considered negative. It enables us also to reduce subtraction of vectors to addition. For if we term the operation denoted by AB-DC a subtraction of the vectors A B and DC, since DC+CD = 0, the operation indicated amounts to adding the vectors A B and C D, or to A B C D. Hence, to subtract two vectors, we reverse the sense of one of them and add.

The result A G + GA=0 can at once be extended to any number of points lying on a straight line. Thus, if P Q R S T U V be a set of such points—

P Q + Q R + RS + ST + TU + UV + VP = 0. For starting from P and taking in succession the steps indicated, we obviously come back to P, or have performed an operation whose result is equivalent to zero, or to remaining where we started.

§ 4. The Addition of Vectors obeys the Commutative

Law.

We can now prove that the commutative law holds for our extended addition (see p. 5). First, we can show that any two successive steps may be interchanged. Consider four successive steps, A B, B C, C D, and D E. If at в instead of taking the step BC we took a step BH equal to C D in magnitude, sense and direction, we could then get from H to D by taking the step HD. Now let B D be joined; then in the triangles B H D, D C B the angles at B and D are equal, because they are formed by the straight line D falling on two parallel lines B H

and CD; also the side B D is common, and в H is equal to C D. Hence it follows (see p. 73) that these triangles are of the same shape and size, or H D is equal to BC; and again the angles B D H and D B C are equal, or H D and B C are parallel. Thus the step H D is equal to the step B C in direction, magnitude and sense. We have then from the two methods of reaching D from B,

BCCD = BD = BH + HD

by what we have just proved.

= CD + BC

B

H

FIG. 58.

Hence any two successive steps may be interchanged. By precisely the same reasoning as we have used on p. 11 we can show that if we may interchange any two successive steps of our zigzag we may interchange any two steps whatever by a series of changes of successive steps; that is, the order in which vectors are added is indifferent.

The importance of the geometry of vectors arises from the fact that many physical quantities can be represented as directed steps. We shall see in the succeeding chapter that velocities and accelerations are quantities of this character.

§ 5. On Methods of Determining Position in a Plane. It has been remarked (see p. 99) that scalar quantities may be treated as steps measured along a

straight line. In this case we only require one point on this line to be given, and we can determine the relative position of any other by merely stating the magnitude of the intervening step. A line is occasionally spoken of as being a space of one dimension; in one-dimensioned space one point suffices to determine the relative position of all others.

When we consider however position in a plane, in order to determine the whereabouts of a point p with regard to another A we require to know not only the magnitude but the direction of the step A P. Hence what scalar steps are to one-dimensioned space, that

are vector steps to plane space. In order to determine the direction of a step AP we must know at least one other point в in the plane. Space which requires two points to determine the position of a third is usually termed space of two dimensions. There are various methods in general use by which position in two-dimensioned space is determined. We shall mention a few of them, confining our remarks however to the plane, or to space of two dimensions which is of the same shape on both sides.

(a) We may measure the distances between A and P and between B and P. If these distances are of