5,6] 



NUMBER. 



6. Geometric Addition. Instead of the point representing 

 the complex number ai + fij + <yk, we may fix our attention upon 





FIG. 1. 



the line drawn from the origin to the representative point. This 

 line is a geometrical magnitude, which is completely specified only 

 when its direction as well as its length is given. Such quantities 

 are called vectors, the name arising from the significance of the 

 operation of carrying a point from one end of the line to the other. 

 Quantities which do not involve the idea of direction, and are 

 completely specified by a single number, are distinguished by the 

 name scalars, the name arising from the possibility of their repre- 

 sentation upon a linear scale*. To specify the direction of a vector 

 we must give two angular coordinates, which together with its 

 length make three data. We may otherwise specify the vector 

 symmetrically by giving its projections on three given mutually 

 perpendicular axes. By projection on, resolved part or component 

 along a line, we mean the product of the length of the vector by 

 the cosine of the angle included between the direction of the 

 vector and the positive direction of the line. If the angle is acute, 

 the projection is positive, if obtuse, negative. In particular the 

 projections of the vector on the axes of i, j, k, are the coefficients 

 of i, j, k, in the representation of the vector by the complex 

 number. It follows from the definition of addition of complex 

 numbers that to add two vectors means to find a vector whose 

 components are the sums of the corresponding components of the 

 two given vectors. This vector may be described geometrically as 

 * If real. Complex scalars may also be used. 



