VECTOR ANALYSIS. 19 



When the vectors to be added are all parallel to the same straight line, 

 geometrical addition reduces to algebraic ; when they have all the same 

 direction, geometrical addition like algebraic reduces to arithmetical. 



It may easily be shown that the value of a sum is not affected 

 by changing the order of two consecutive terms, and therefore that 

 it is not affected by any change in the order of the terms. Again, it 

 is evident from the definition that the value of a sum is not altered 

 by uniting any of its terms in brackets, as a + [/3+y] + etc., which is 

 in effect to substitute the sum of the terms enclosed for the terms 

 themselves among the vectors to be added. In other words, the 

 commutative and associative principles of arithmetical and algebraic 

 addition hold true of geometrical addition. 



7. Def. A vector is said to be subtracted when it is added after 

 reversal of direction. This is indicated by the use of the sign 

 instead of + . 



8. It is easily shown that the distributive principle of arithmetical 

 and algebraic multiplication applies to the multiplication of sums of 

 vectors by scalars or sums of scalars, i.e., 



+ m/3+ n/3 + etc. 

 + etc. 



9. Vector Equations. If we have equations between sums and 

 differences of vectors, we may transpose terms in them, multiply or 

 divide by any scalar, and add or subtract the equations, precisely as 

 in the case of the equations of ordinary algebra. Hence, if we have 

 several such equations containing known and unknown vectors, the 

 processes of elimination and reduction by which the unknown vectors 

 may be expressed in terms of the known are precisely the same, and 

 subject to the same limitations, as if the letters representing vectors 

 represented scalars. This will be evident if we consider that in the 

 multiplications incident to elimination in the supposed scalar equations 

 the multipliers are the coefficients of the unknown quantities, or 

 functions of these coefficients, and that such multiplications may be 

 applied to the vector equations, since the coefficients are scalars. 



10. Linear relation of four vectors, Coordinates. If a, /3, and y 

 are any given vectors not parallel to the same plane, any other vector 

 p may be expressed in the form 



If a, /3, and y are unit vectors, a, b, and c are the ordinary scalar 

 components of p parallel to a, /3, and y. If p = OP, (a, /3, y being 

 unit vectors), a, b, and c are the cartesian coordinates of the point P 

 referred to axes through O parallel to a, /3, and y. When the values 

 of these scalars are given, p is said to be given in terms of a, /3, and y. 





