14 GEOMETRY OF VECTORS. [CHAP. II. 



II. Consider the more general case where the vectors are not 

 parallel to a plane. Let r l} r 2 , ...r n be the magnitudes of the 

 vectors, and call any one of these numbers r. Let I, m, n be the 

 cosines of the angles which the line representing this vector in 

 direction and sense makes with the axes of Ox, Oy, Oz. Then this 

 vector may be resolved into rl, rm, rn parallel to the lines Ox, Oy, 

 Oz, and the whole set of vectors is equivalent to a vector whose 

 resolved parts parallel to the axes are X, Y, Z, where X = %rl, 

 Y = Xrm, Z 2rn, the summations extending to all the vectors of 

 the set. The resultant is therefore a vector whose magnitude, R, 

 is the numerical value of *J(X Z + Y 2 + Z*), and such that the line 

 representing it in direction and sense makes with the axes Ox, Oy, 

 Oz angles whose cosines are X/R, Y/R, Z/R. 



11. Vectors equivalent to zero. When the magnitude of 

 the resultant of any set of vectors is zero the set of vectors is said 

 to be equivalent to zero. Thus two equal vectors parallel to the 

 same line, and in opposite senses, are equivalent to zero. 



It is clear that the sum of the resolved parts, in any direction, 

 of a set of vectors equivalent to zero is equal to zero. 



Again vectors parallel and proportional to the sides of a 

 polygon, and with senses determined by the order of the corners 

 when a point travels round the polygon, are equivalent to zero. 



This last statement enables us to do away with the restriction 

 (Art. 8) that in the resolution of a vector into components 

 parallel to the sides of a polygon not more than two sides of the 

 polygon may meet in a point. 



12. Centroids. Although not immediately connected with the subject 

 of this Chapter, it is convenient here to introduce the definitions of the 

 centroids of figures. 



Consider in the first place a figure consisting of isolated points A lt A 2 ,...A n . 

 Let any plane be drawn, and let x lt x^...x n be the distances of the points 

 from the plane, a positive sign being given to the tfa of the points on one side 

 of the plane, and a negative sign to the ^s of the points on the other side of the 

 plane. Then there will be a parallel plane whose distance from this plane is 



- (x 1 +x 2 +... + x n ) ; the sign of this quantity determines the relative situation 



of the planes. We may say that the distance of this plane from the plane 

 first drawn is the average distance of the points from the plane first drawn. 



