CHAPTER IV ON POINTS AND VECTORS IN SPACE SECTION 1 On the Mean Point 33°. Definition.-If the sum, Za, of m coinitial vectors, coplanar or non-coplanar, =a2 ΟΑ = α1, OA2 OAm = am be divided by their number, m, the resulting vector, is the Simple Mean of those m vectors, and its term, M, is the Mean Point of the system of points, A1, A2... Am If we are given such a system of points, a1, a2 and also a system of scalars, P1, P2 D Pn; the vector an is the Complex Mean of those n vectors, and its term, C, is the Centre of Gravity, or Barycentre, of the system of points, A1, A2 An, considered as loaded with the given weights, P1, P2 Pn (Hamilton). . 34°. The position of the mean point depends upon the configuration of the system, and is independent of the position of the arbitrary origin, O. For, let C be the mean point of a given system, A, An, with respect to an assumed origin, O; and let C' be the mean point of the same system with respect to another assumed origin O' ; let O'A1, O'A2, &c., be represented by a'1, a'2, &c., and let γ be the mean vector with respect to O, y' the mean vector with respect to O'. Then, But the equal vectors, OC, OC', have a common origin; therefore they must have a common term, or, C = C'. _The position of the mean point has not been altered, therefore, by selecting O' instead of O as the origin of the system. Pms 35°. The sum of the m vectors, P1, P2 drawn from the mean point to every point of the system, is zero. Conversely, if C be such a point that the sum of the vectors drawn from it to each and every point of a given system is zero; then C is the mean point of the system. 36°. If any system of points, together with its mean point, be projected by parallel ordinates upon any plane, the projection of the mean point is the mean point of the projected system. ... An Let A', .. A'n be the projections of the points A, Find M', the mean point of the projected system; draw MM', and let ... MAI MAN Pn; M'A', M'A', p'... P'n Therefore MM' is parallel to έ, that is, to the other ordinates. Therefore M' is the projection of M, the mean point. of the mean point is the mean of the ordinates of the system. SECTION 2 Linear Equations connecting four Non-coplanar Vectors 37°. It has been shown, 27°, that if three vectors, a, ß, y, cannot be FIG. 7. For equated to zero unless all three coefficients vanish. ha + lẞ=pp, some vector in the plane OAB, fig. 7, and pp + my = qέ, some vector in the plane containing i.e., some plane different from OAB, OBC, and OCA. the expression, ha + lẞ + my, represents some fourth and Y, Hence vector, h n .. n n If we take OA'= a, OB'= B, OC'= n complete the parallelopiped OA"B"C", we determine a point D, such that, OD=OC"+C"D= OC" + OC'= OA'+ OB'+ OC' Hence, since a, ẞ, y may be any actual vectors, and since h, l, m may have any values whatever, the sum of the three coinitial edges of a parallelopiped is the internal coinitial diagonal. 38°. If ha + lẞ + my + nd = o, what must be the relation between the scalars in order that the point D may be situated in the fourth given plane ABC; or, in other words, what is the condition of coplanarity of the four points, A, B, C, D? If D lies in the plane ABC, the vectors DA, DB, DC are coplanar, and are consequently, 27°, connected together by an equation of the form or, or, But pa + qẞ + ry - (p + q + r) d = 0. ha + B + my + nô=0. Hence, eliminating 8 from the last two equations, {h (p + q + r) + np} a + {l (p + q + r) + ng} ß Now, if the coefficients have an actual value, a, ß, y are coplanar. But, by hypothesis, a, ß, y are not coplanar. Therefore the coefficients have not an actual value, and must be equated to zero. Therefore, |