Electromagnetic Wave-surface. 401 
where A, B,..., are any vectors, which, if m in number, may 
be represented, since their sum is zero, by the n sides of a 
polygon. Let A,, A,, A; be the three ordinary scalar com- 
ponents of A referred to any set of three rectangular axes, 
and similarly for the other vectors. This notation saves mul- 
tiplication of letters. Then the above equation stands for the 
three scalar equations 
A,+B,+0C,+D,+. 7.05 
A,+B,+C,+ D.+. ° =i 
A;+B;+C;+D3;+.. _= 0: 
The — sign before a vector simply reverses its direction— 
that is, negatives its three components. 
According to the above, if 2, 7, & be rectangular vectors of 
unit length, we have 
—— 1A, +jA, -- kA3 ° . e ° ° ° (1) 
&c. ; if Ay, A,, A; be the components of A referred to the 
axes of 1,7, k. That is, A is the sum of the three vectors 
tAy, jAs, kA3, of lengths A,, Ay, A; parallel to 2, 7, & respec- 
tively. 
Scalar Product.—We define AB thus, 
AB = A,B, + A,B, + A3Bs, ° ° : . (2) 
and call it the scalar product of the vectors A and B. Its 
magnitude is that of A x that of B x the cosine of the angle 
between them. Thus, by (1) and (2), 
i — A, Ae a A,;=Ak 3 
and in general, N being any unit vector, AN is the scalar 
component of A parallel to N, or, briefly, the N component 
of A. Similarly, 
Pah OPS Pa 
because i and 7 are parallel and of length unity, &e. And 
g—Oy 7e=05 t=O, 
because 7 and 7, for instance, are perpendicular. Notice that 
AB=BA. 
We have also Sn a eae 
mi ee mee 
and 
+ or Ate t= 45 = be 
Thus A has the same direction as A; its length is the 
reciprocal of that of A. 
