6 I. KINEMATICS OF A POINT. LAWS OF MOTION. 



The nature of the construction shows that the resultant is inde- 

 pendent of the order of taking the components. 



Since a negative quantity is denned as that which added to a 

 given positive quantity produces zero, the negative of A 13 must be 

 BA, for hy the above rule, 



A+ AB = B, 



therefore A + AB 



The coordinates of BA are also the negatives of AB. The 

 scalar length of a vector is called by Hamilton its tensor, so that 

 the tensor of the negative of a vector is the same as that of the 

 vector itself. 



It is evident from the definition of a vector that the projection 

 of the sum of two vectors on any direction is the algebraic sum of 

 the projections of the components. Projecting on the three directions 

 of the coordinate -axes, and distinguishing the projections of the 

 components by suffixes, we have for the projections of the resultant, 



S x = Six + Sz x , 



S = Sl + S 2 , 



S 2 = (S ix -f S 2 *) 2 + (Sly + $2y) 2 + (Siz + S2*) 2 , 



and for the sum of any number of vectors, 



8) S 2 =(Is,) 2 +(I Sj ,) 2 +(I S2 ) 2 . 



We may easily find an expression for the projection of any 

 vector ~s upon any direction, which is given by its direction cosines, 

 cos A, cos p, cos v. We have for the angle # between two lines 

 whose direction cosines are cos I, cos p, cos v, cos Z r , cos ^ , cos i/, 



cos & = cos I cos ^ + cos 11 cos ^ + cos v cos v\ 

 but by 6), we have for "s, 



s s s, 



cosA' = i cos//= > cosi/= i 



s s s 



BO that 



9) s cos # = s x cos A + s y cos 11 -f s z cos Vj 



which is the expression for the projection. Taking for the direction 

 of projection the direction of the vector itself, this becomes equation 7). 



