INTRODUCTION. SECTION (I). SCALARS AND VECTORS. 1. THE different quantities dealt with in Mathematics and Physics, e.g. numbers, lines, lengths, velocities, forces, time, temperature, etc. have been divided into two groups according as they are or are not related to a definite direction in space. Quantities such as Numbers, Time, Mass, Length, Energy and Temperature have no such relation and may be completely specified by giving (i) the kind of quantity, (ii) how much there is of it. Thus a length of 3 feet, a time interval of 10 seconds, a temperature of 98° Centigrade, completely determine the length, time and temperature respectively, without any reference to direction in space. 2. If u denote the unit of such a quantity and A the number of units contained, the quantity is completely determined by the expression Au. This permits of a geometrical representation, for if we agree to represent u, the unit, by a line of definite length, then Au would be represented by a line A times as long as the first line. Thus, if u denote a degree Centigrade and we represent this by a line of length 1 inch, then 65° would be represented by a line 6.5 inches in length and 212° by a line 21.2 inches in length. This process is called representing the quantities to 'scale' by lengths, and the quantities themselves are called Scalar Quantities. (From the Latin scalae-a ladderdivided into equal parts by the rungs.) 3. Definition. A quantity which has no relation to a definite direction in space, or which is considered apart from such direction, is called a Scalar-Quantity or simply a Scalar. 4. Definition. A quantity which is related to a definite direction in space is called a Vector-Quantity or simply a Vector. The displacement of a point, velocity, momentum, acceleration, force, rotation about an axis, an electric current, are examples of vectors; none of them can be considered as completely specified without some reference to direction. Α Fig. 1. Β ́ 5. Displacement. If a point is moved or displaced from A to B, then the amount of the displacement is the length of the line AB, say 1 foot. If all we know about the point B is that it may be found by displacing A 1 foot, then we do not know the position of B relative to A. All we know is that B is somewhere on the surface of a sphere having A as centre and a radius of 1 foot. If, however, we know also the direction of the displacement, say along a line drawn East and West through A, then B may occupy a position either 1 foot East or 1 foot West of A. To fix which of these it is, we must say, in addition, towards the East or towards the West. This third thing necessary to fix the position of B relative to A is called the sense of the displacement in the given direction. Displacement is thus a Vector-Quantity and requires for its complete determination the magnitude, direction and sense to be given. The magnitude or amount of the displacement is a scalar. 6. A vector-quantity can be geometrically represented by a line, if (1) the length of the line represents to scale the magnitude of the quantity, (2) the line be placed in the proper direction, The sense is usually indicated by an arrowhead on the line. The line itself with its direction and sense is called a vector. A vector must not be understood to occupy any definite position in space. The following illustration will help to make this clear. Suppose a number of points to move in parallel straight lines with the same speed and in the same sense, then the same vector will represent the displacement of each point in the same time and therefore the position we suppose the vector to be placed in is immaterial. The wind and the motion of translation of a rigid body are good examples of this. In the case of the wind, we cannot say that it has any particular position; it blows over any small area with the same speed at every point. The wind has magnitude, direction and sense, and that is all. 7. Some physical quantities require for their specification not only magnitude, direction, and sense to be given, but also position. Thus a force is not completely determined unless the line of action is given as well as the magnitude, direction and sense. As another example of such quantities, consider the motion of a flywheel or any spinning body. To describe the motion of such a body we must state about what axis of the body it is spinning; it is not sufficient to give its direction, for the motion about any axis is quite different to what it would be about a parallel axis. The magnitude of the spin is given by the angle turned through per second, while the sense of the spin is fixed when we know whether the rotation is right or left handed. Forces, rotations and spins are thus to be considered as vector quantities having definite position. Any quantity whether scalar or vector, considered as occupying a definite position in space, is said to be localized. Thus the mass of a body in a given position is a localized scalar, a force acting on a body at a definite point is a localized vector. 8. Definition. A localized vector is called a Rotor (Clifford). 9. All quantities used by Euclid and those used in ordinary Algebra are scalars, and these branches of Mathematics treat fully of the Mathematics of Scalars. There is also a Mathematics of Vectors which forms the subject matter of this book. It will be only necessary to consider two operations, viz. those of Addition and Multiplication. The rules guiding these differ essentially from those for scalars. Their definitions, though to a certain extent arbitrary, are chosen in such a manner that the results obtained express in a concise and simple form facts met with in Dynamics. and Physics. To compare these rules with those used in ordinary Algebra and to emphasize the points wherein they differ, it is desirable to briefly consider the laws which govern ordinary Algebra. SECTION (II). THE LAWS OF ALGEBRA. 10. Addition. Let a, b, c, d, ... be algebraical quantities, then for Addition there are two laws. The first law says that in forming the sum of a and b we may either add b to a, or a to b; in symbols |