20 VECTOR ANALYSIS. 



It is generally in this way that the value of a vector is specified, viz., 

 in terms of three known vectors. For such purposes of reference, a 

 system of three mutually perpendicular vectors has certain evident 

 advantages. 



11. Normal systems of unit vectors. The letters i, j, k are appro- 

 priated to the designation of a normal system of unit vectors, i.e., 

 three unit vectors, each of which is at right angles to the other two 

 and determined in direction by them in a perfectly definite manner. 

 We shall always suppose that k is on the side of the i-j plane on 

 which a rotation from i to j (through one right angle) appears 

 counter-clockwise. In other words, the directions of i, j, and k are 

 to be so determined that if they be turned (remaining rigidly con- 

 nected with each other) so that i points to the east, and j to the north, 

 k will point upward. When rectangular axes of X, Y, and Z are 

 employed, their directions will be conformed to a similar condition, 

 and i, j, k (when the contrary is not stated) will be supposed parallel 

 to these axes respectively. We may have occasion to use more than 

 one such system of unit vectors, just as we may use more than one 

 system of coordinate axes. In such cases, the different systems may 

 be distinguished by accents or otherwise. 



12. Numerical computation of a geometrical sum. If 



etc., 

 then 



p + 0-+ etc. = (a + a' + etc.) a + (b + b' + etc.) /3 + (c + c' + etc.) y, 



i.e., the coefficients by which a geometrical sum is expressed in terms 

 of three vectors are the sums of the coefficients by which the separate 

 terms of the geometrical sum are expressed in terms of the same 

 three vectors. 



Direct and Skew Products of Vectors. 



13. Def. The direct product of a and ft (written a. ft) is the scalar 

 quantity obtained by multiplying the product of their magnitudes 

 by the cosine of the angle made by their directions. 



14. Def. The skew product of a and ft (written ax ft) is a vector 

 function of a and ft. Its magnitude is obtained by multiplying the 

 product of the magnitudes of a and ft by the sine of the angle made 

 by their directions. Its direction is at right angles to a and ft, and 

 on that side of the plane containing a and ft (supposed drawn from 

 a common origin) on which a rotation from a to ft through an arc 

 of less than 180 appears counter-clockwise. 



The direction of ax ft may also be defined as that in which an 

 ordinary screw advances as it turns so as to carry a toward ft. 



