where y is plus or minus, according as the vectors have the same or opposite directions. y may be called the geometric quotient, and is a real number, plus or minus, expressing numerically the ratio of the vector lengths. This quotient of parallel vectors, which may be positive or negative, whole, fractional or incommensurable, but which is always real, is called a Scalar, because it may be always found by the actual comparison of the parallel vectors with a parallel right line as a scale. It is to be observed that tensors are pure numbers, or signless numbers, operating only metrically on the lengths of the vectors of which they are coefficients: while scalars are sign-bearing numbers, or the reals of Algebra, and are combined with each other by the ordinary rules of Algebra; they may be regarded as the product of tensors and the signs of direction. Thus, let Then Taα. - b, a=aUa. If we increase the length of a by the factor b, b is a tensor, but the tensor of the resulting vector is ba. If we operate with b is not a tensor, for a is not only stretched but also reversed; the tensor of the resulting vector is as before ba; in other words, direction does not enter into the conception of a tensor. As the product of a sign and a tensor, -b is a scalar. The operation of taking the scalar terms of an expression is indicated by the symbol S. Thus, if c be any real algebraic quantity, S(-ba Ua + c) = c, for - ba Ua is a vector, and the only scalar term in the expression is c. 9. It is evident from Art. 7 that if a, b, c are scalar coefficients, and a any vector, we have then, A'B' being drawn parallel to AB and B'c' to BC, or the distributive law holds good for the multiplication of scalar and vector quantities. a ẞ cannot be zero, since no amount of transference in a direction not parallel to a can affect a. Hence, if na + mẞ = 0, since a and ẞ are entirely independent of each other, we must have A Three or more vectors may, however, neutralize each other. where a, ß, y, 8, ...... 11. Examples. 1. The right lines joining the extremities of equal and parallel Hence (Art. 2), y = ß, and вo is parallel and equal to DA. 2. The diagonals of a parallelogram bisect each other. In Fig. 8 we have also BD = OA = OP + PA; BD = BP + PD ; .. OPPA = BP + PD. But, op and PD being in the same right line, 3. If two triangles, having two sides of the one proportional to two sides of the other, be joined at one angle so as to have their homologous sides parallel, the remaining sides shall be in a straight line. Let (Fig. 9) AB = a, AE = ß. Then, by condition, DC = xa, DB= xß. Now Fig. 9. D Hence (Art. 2), в being a common point, CB and BE are one and the same right line. 4. If two right lines join the alternate extremities of two parallels, the line joining their centers is half the difference of the parallels. 5. The medials of a triangle meet in a point and trisect each BP+PA BA = 20D = 2 (OP + PD). But BP and PD, as also OP and PA, lie in the same direction, and therefore BP 2 PD and PA=20P. Hence the medials OA and DB trisect each other. CP = CB + BP = 3 (2a + ẞ) — 2a = } (ẞ − a), Hence PE and CP are in the same straight line, or the medials meet in a point. 6. In any quadrilateral, plane or gauche, the bisectors of opposite sides bisect each other. Fig. 12. B We will first find a value for OP (Fig. 12) under the supposition that P is the middle point of GE. We shall then find a value for Op, under the supposition that P is the middle point of FH. If these expressions prove to be identical, these middle points must coincide. In this, as in many other problems, the solution depends upon reaching the same point by different routes and comparing the results. P F E H A |