CHAPTER I.

SECTION (I). THE ADDITION OF VECTORS.

25. Definition. A Vector is a Geometrical Quantity which is related to a definite direction in space.

At first only purely geometrical vectors (line vectors) will be considered, in applications however the vector may represent any physical quantity which has magnitude, direction, and sense.

The magnitude of a Vector is then a length, but the magnitude of a vector-quantity may be that of any quantity whatever, a force, a current, a mere number, etc.

This magnitude was called by Hamilton the Tensor of the vector.

Unit Vector. A vector whose length is the unit of length is called a unit vector.

Ort. A vector whose length represents the number 1 is called an 'ort' (short for orientation, Heaviside) and indicates simply direction.

26. The specification of a vector requires the magnitude, direction and sense to be given. The magnitude is given by the length of a line, its direction by the direction of, and its sense by an arrowhead on, the line.

A

Thus in the figure, AB is a vector of magnitude given by the length of AB, of direction shewn and of sense from A to B.

Fig. 13.

B

A vector has no definite position in space, and may therefore (without alteration) be moved parallel to itself.

All vectors which have the same magnitude, direction and sense are equal. Though a vector remains unaltered if moved parallel to itself, it would be another vector altogether if its direction, sense or magnitude were changed.

27. Notation. Vectors will be denoted by small Greek letters and with few exceptions (in the case of angles) small Greek letters will always denote vectors.

28. Position Vector. The difference in position of two points A and B in space is known when the step AB is given in magnitude, direction and sense. This is a

vector and tells us that if we start from A and travel in the direction and sense indicated for a distance AB, we shall arrive at B. Hence given a point O and a step or vector 04, the position of A is determined. This vector is called the position vector of A relative to O as origin.

29. For steps in a line we had the equation

whatever the relative positions of A, B, and C.

This may be extended to the case where the points are not in a line, for the result of taking the step AB and then the step BC is exactly the same as taking the single step AC. If a point moves from A to B and then from B to C the final position of the point is as if it had moved direct to C. In this sense we may

say that the sum of the steps AB and BC is equivalent to the step AC. These steps being vectors we may define the sum of two vectors as follows:

:

Definition. If two vectors are placed so that the beginning of the second coincides with the end of the first, then the vector from the beginning of the first to the end of the second is called the sum of the two vectors.

Denoting the steps AB, BC, and AC by a, B, and y we have the vector equation

a + B

=

30. We see at once that the Commutative Law holds, viz.:

a + B = B+ a.

For BC and AB (see Fig. 14) may be moved parallel to themselves to the positions AB' and B'C and

AB' + B'C = AC.

31. Since we can add two vectors, any number can be added by a continuation of the above process, the rule being:

α

γ

To add a number of vectors, place the first anywhere, the beginning of the second to the end of the first, the beginning of the third to the end of the second and so on, then the vector from the 8 beginning of the first to the end of the last is the sum of the given

Fig. 15.

vectors.

The sum of a number of vectors is often called the Resultant Vector, and in relation to this resultant the other vectors are called components. In the figure, σ is the sum or resultant vector of the vectors, a, ẞ, y and 8; and we write

σ = a+B+y+8.

32. The rule for the addition of vectors does not imply that the vectors are in a plane or, what amounts to

the same thing, in parallel planes. If they are not coplanar, the polygon formed by placing the vectors end to end will lie in space and not in a plane, but the above rule holds.

Definition. This polygon whether plane or skew is called the Vector Polygon.

33. The law of association holds. From the figure it is seen that

34. If the Vector Polygon is closed, the sum of the vectors vanishes, and conversely, if the sum of the vectors vanishes the Vector Polygon is closed.

In the figure a, B, y, 8 are vectors and the vector

polygon ABCD is closed, and therefore

a+B+y+8= 0.

If it is given that

a+B+y+8=0,

then the polygon is closed.

35. Like Vectors. Since vectors can be moved parallel to themselves, all vectors which have the same direction can be brought to lie in the same line and are called like vectors, like vectors therefore can only differ in magnitude and sense. The vector polygon for like vectors is a straight line in the direction of the vectors, and the sum has therefore the same direction and its magnitude is the sum of the magnitudes of the given vectors. Hence like vectors are added by adding their magnitudes.

If n equal vectors a are added, the sum will be a like vector of n times the length of a. It is as in Algebra denoted by na; this leads to the following definition.

Definition. Multiplying a vector by a number means multiplying its length by that number.

1

Conversely any vector a is the sum of n equal vectors, each of length th of a; hence dividing a vector by a number means dividing its length by that number.

n

These definitions, although first obtained for whole numbers n, retain a meaning when n is any number, positive or negative, whole or fractional or incommensurable.

Any vector can be expressed as a multiple of a like vector, or if a and B are like vectors

B = ma,

where m is the ratio of the lengths of ß and a, and hence a number.

36. A similar relation holds also if the vectors are not line vectors but any vector quantities. Thus if be an ort giving the direction and sense of a and if a be the length of a, then

a = ai,

and hence the ort is given by

a/a = i.