16 GEOMETRY OF VECTORS. [CHAP. II. 



14. Localised Vectors. The vectors we have so far con- 

 sidered have no relation to any particular point, they are equally 

 well represented by lines drawn from any point, and they have no 

 relation to any particular line, they are equally well represented 

 by segments of all lines parallel to their direction. They may be 

 called unlocalised vectors. But it is often important to consider 

 objects of mathematical reasoning which, in other respects, have 

 the properties of vectors, but which have relations with particular 

 points or particular lines. 



A vector localised at a point is an object of mathematical 

 reasoning which, like an unlocalised vector, is defined by its 

 magnitude, direction, and sense, and also by a point and by a rule 

 of equivalence, viz. : two sets of vectors localised at the same 

 point are equivalent if two sets of unlocalised vectors with the 

 same magnitudes, directions, and senses are equivalent. 



There is in general no rule of equivalence for vectors localised 

 at different points. 



A vector localised in a line is a vector localised at any point in 

 a particular line, which is in the direction of the vector, with the 

 additional rules of equivalence that two vectors localised in the 

 same line are equivalent if they have the same magnitude and the 

 same sense, and that two vectors localised in lines which meet are 

 equivalent to a single vector localised in a line. 



All the constructions in the previous Articles apply to vectors 

 localised at points and to vectors localised in lines, provided all 

 components and resultants are localised at the proper points or in 

 the proper lines. In particular a vector localised at a point is 

 equivalent to components (or resolved parts) of the same mag- 

 nitudes, directions, and senses as if it were unlocalised, provided 

 these components and resolved parts are localised at the same 

 point; also a vector localised in a line is equivalent to components 

 (or resolved parts) of the same magnitudes, directions, and senses 

 as if it were unlocalised, provided these components and resolved 

 parts are localised in lines which meet in a point on the line of 

 the resultant. 



Thus a vector localised at may be represented (as in Fig. 11) 

 by a line OP, and is equivalent to vectors localised at and 

 represented by lines OH, OK, OM ; and a vector localised in the 



