22 VECTOR ANALYSIS. 



and, in general, direct and skew products of sums of vectors may be 

 expanded precisely as the products of sums in algebra, except that 

 in skew products the order of the factors must not be changed 

 without compensation in the sign of the term. If any of the terms 

 in the factors have negative signs, the signs of the expanded product 

 (when there is no change in the order of the factors) will be deter- 

 mined by the same rules as in algebra. It is on account of this 

 analogy with algebraic products that these functions of vectors are 

 called products and that other terms relating to multiplication are 

 applied to them. 



21. Numerical calculation of direct and skew products. The 

 properties demonstrated in the last two paragraphs (which may be 

 briefly expressed by saying that the operations of direct and skew 

 multiplication are distributive) afford the rule for the numerical 

 calculation of a direct product, or of the components of a skew 

 product, when the rectangular components of the factors are given 

 numerically. In fact, if 



and 3 = x 



and ax/5 = (yz' - zy')i + (zx f - xz')j + (xy' - yx')k. 



22. Representation of the area of a parallelogram by a skew 

 product. It will be easily seen that ax/3 represents in magnitude 

 the area of the parallelogram of which a and /3 (supposed drawn 

 from a common origin) are the sides, and that it represents in direc- 

 tion the normal to the plane of the parallelogram on the side on 

 which the rotation from a toward /3 appears counter-clockwise. 



23. Representation of the volume of a parallelopiped by a triple 

 product. It will also be seen that aX/3.y* represents in numerical 

 value the volume of the parallelopiped of which a, /3, and y (supposed 

 drawn from a common origin) are the edges, and that the value of 

 the expression is positive or negative according as y lies on the side 

 of the plane of a and /3 on which the rotation from a to /3 appears 

 counter-clockwise, or on the opposite side. 



24. Hence, 



= /3.yXa= /3xa.y= yX/3.a= aXy./3 

 = y./3xa= a.yX/3= /3.aXy. 



It will be observed that all the products of this type, which can be 

 made with three given vectors, are the same in numerical value, and 



* Since the sign x is only used between vectors, the skew multiplication in expressions 

 of this kind is evidently to be performed first. In other words, the above expression 

 must be interpreted as [ax/3].-y. 



