VECTOR ANALYSIS. 23 



that any two such products are of the same or opposite character in 

 respect to sign, according as the cyclic order of the letters is the same 

 or different. The product vanishes when two of the vectors are 

 parallel to the same line, or when the three are parallel to the same 

 plane. 



This kind of product may be called the scalar product of the three 

 vectors. There are two other kinds of products of three vectors, both 

 of which are vectors, viz., products of the type (a./3)y or y(a./3), and 

 products of the type ax[/8xy] or [yx/3]xa. 



25. 



From these equations, which follow immediately from those of 

 No. 17, the propositions of the last section might have been derived, 

 viz., by substituting for a, /3, and y, respectively, expressions of the 

 form xi + yj+zk, x'i+y'j+z'k, and x"i+y"j+z"k.* Such a method, 

 which may be called expansion in terms of i, j, and k, will on many 

 occasions afford very simple, although perhaps lengthy, demonstrations. 

 26. Triple products containing only two different letters. The 

 significance and the relations of (a.a)/3, (a./3)a, and ax[aX/3] will 

 be most evident, if we consider /3 as made up of two components, 

 ft and /3", respectively parallel and perpendicular to a. Then 



(a . /5) a = (a . (3') a = (a . a)/3', 



Hence, ax[aX/3] = (a./3)a (a.a)/3. 



27. General relation of the vector products of three factors. In the 

 triple product ax[/3xy] we may set 



a = Z/3 + ray + 71/3 X y, 

 unless /3 and y have the same direction. Then 



But /3.y + my.y = a.y, and / 



Therefore ax[^Xy] = (a.y)/3 (a./3)y, 



which is evidently true, when /3 and y have the same directions. It 

 may also be written 



* The student who is familiar with the nature of determinants will not fail to observe 

 that the triple product a.pxy is the determinant formed by the nine rectangular 

 components of a, /3, and 7, nor that the rectangular components of ax/S are determinants 

 of the second order formed from the components of a and /S. (See the last equation of 

 No. 21.) 



