VECTOR ANALYSIS. 25 



the X-, Y-, and Z-components of X and adding. Hence the second 

 equation may be regarded as the most general form of a scalar equa- 

 tion of the first degree in a, 6, c, d, etc., which can be derived from 

 the original vector equation or its equivalent three scalar equations. 

 If we wish to have two of the scalars, as b and c, disappear, we have 

 only to choose for X a vector perpendicular to /3 and y. Such a 

 vector is /3xy. We thus obtain 



37. Relations of fowr vectors. By this method of elimination we 

 may find the values of the coefficients a, 6, and c in the equation 



p = oa + 6/3+cy, (1) 



by which any vector p is expressed in terms of three others. (See 

 No. 10.) If we multiply directly by /3xy, yXa, and aX/3, we obtain 



p./3xy = aa./3xy, p.yXa = 6/3.yXa, p.aX/3 = cy.aX/3; (2) 

 whence 



- 



By substitution of these values, we obtain the identical equation, 



(a./3xy)p = (p./3xy)a + (/o.yXa) ft + (p.aX/3)y. (4) 



(Compare No. 31.) If we wish the four vectors to appear symmetri- 

 cally in the equation we may write 



(a.xy)p - (0.yxp)a + (y.pXa)/3 - (p.ax)y = 0. (5) 



If we wish to express p as a sum of vectors having directions 

 perpendicular to the planes of a and /3, of /3 and y, and of y and a, 

 we may write 



p = e/3xy+/yXa+#ax/3. (6) 



To obtain the values of e, f, g, we multiply directly by a, by /8, and 

 by y. This gives 



p- a f- p-P _ 



- -> ~ 



Substituting these values we obtain the identical equation 



(u./3xy) p = (p.a) /3xy + (p./3) yXa + (p.y) ax/8. (8) 



(Compare No. 32.) 



38. Reciprocal systems of vectors. The results of the preceding 

 section may be more compactly expressed if we use the abbreviations 



_. _. 



" 3 P ~' ~ 



The identical equations (4) and (8) of the preceding number thus 



become 



(2) 



(3) 



