14, 15] LOCALISED VECTORS. 17 



line OP, having the same magnitude and sense, is equivalent to 

 vectors localised in any three lines parallel to Ox, Oy, Oz, meeting 

 in a point on OP, and having the magnitudes and senses of OH, 

 OK, OM. 



The differences between the three classes of vectors may be 

 expressed thus : 



A vector (unlocalised) is equivalent to any parallel vector of 

 equal magnitude and like sense. Thus the line representing the 

 vector may be drawn from any point. 



A vector localised in a line is equivalent to any vector of equal 

 magnitude and like sense localised in the same line. The line 

 representing it may be drawn from any point in a particular line, 

 and is a segment of that line. 



A vector localised at a point is not equivalent to any other 

 single vector. The line representing it must be drawn from the 

 point. 



A vector localised in a line is clearly determined by its com- 

 ponents parallel to three given lines and by one point of the line, 

 in particular the line in which it is localised is thereby deter- 

 mined. 



15. Equivalent systems of vectors localised in lines. 



Let two vectors be localised in lines which meet in a point A. 

 They may be localised at the point A. They are then equivalent 

 to a determinate resultant localised at A, and since, by definition, 

 they are equivalent to a vector localised in a line, this line is the 

 line of the resultant through A. 



Let AB, AD be the lines in which the vectors are localised, 

 the order in which the points are 

 named indicating the senses. Let 

 P and Q be their magnitudes; and 

 take the segments AB and AD to 

 be proportional to P and Q. Con- 

 struct a parallelogram A BCD 

 having AB, AD as adjacent sides. 

 Then a vector localised in the line 

 L. 



