MetuHops IN Sotrp ANALYTICS. 
By ArtHuR S. HatTHaway. 
Define the ‘‘vector” [h, k, m] as the carrier of the point (x, y,z) = P, to the 
point (x +h, y+hk, z+m)=Q, and show that the distance and direction cosines 
of the displacement P@ are given by functions of the vector called its tensor and 
re Th, k, m] = 4" + #2 4 m2) =n, ULh, km] =[h/n, k/n, m/n]. 
Interpret the sum [A, k, m] + [W’, ’, mJ] =[h+W,k4+ kh, m+ m’] asa 
resultant displacement, PQ + QR = PR, and the product n[h, k,m] =[nh, nk, nm], 
as a repetition of the displacement. 
’ or ‘‘vectors” whose 
Define the linear functions of g=[2, y,z] as the ‘‘scalars’ 
values or components are linear homogeneous functions of the components of q, 
such as az + by + cz, etc. Hence, fora linear function Fg, F(q+7r) = Fq+ Fr, 
nFq = F (nq). 
Hence, for a bi-linear function Far, F(ag 4+ aq’, br + b’r’) = abFqr + 
ab’ Far’ + a/b q’r + ab’ Fq’r’. 
A special scalar and vector bilinear function of g=[z, y, 2], g’ =[’, y, 2] 
are defined. 
Sqq’= xx’ + yy’ + 2/=Sq'q. 
Vqq= Lye’ — zy, 2x’ — xz, xy/— yx’ | = — Vq’q. 
If © be the angle between the displacements g, g’, these functions are inter- 
preted as, 
Sqq’ = Tq . Tq’. cos®8. TVqq’=Tq.Tq’. sin8; and Vqq’ is a displace- 
ment perpendicular to both g and q’, in the same sense as the axis OZ is per- 
pendicular to OX and OY, 7. e., on one side or the other of the plane XOY. 
The use of this material is illustrated in the following examples: 
Met? 3d, — 1), b= (3, 0, 11) 6 — (850-2), Di (d,7, 11). 
1. Find the lengths and direction cosines of AB, AC, AD. 
Ans. LAB = 3, UAB [4)-3;,-4], ete. 
2. Find cosBAC. Ans. SUABUAC=}$- 
3. Find area of ABC and volume of ABCD. 
Ans. 4 TVABAC=} 185, 4 SADVABAC= — 13. 
4. Find the cosine of the diedral angle C— AB— D. 
Ans. SUVABACUVABAD = mS 
5. Find the sine of the angle between AD and the plane ABC. 
han SU PAB AC ===. 
