$o.] TRANSLATIONS. 2 



47. The process of geometric addition explained in Art. 46 

 for the case of two components is readily extended to the gen- 

 eral case of n components. It thus appears that the succession of 

 any number of translations of a rigid body has for its resultant 

 a single translation whose vector is found by geometrically adding 

 the vectors of the component translations. (Compare Art. 7.) 



48. The order in which vectors are combined, or added, is 

 indifferent for the result. This is directly apparent from a 

 figure in the case of two vectors (Fig. 14). 



For the case of n vectors it follows from 



JL A 



the consideration that any order of the vec- 

 tors can be obtained by repeated interchanges 

 of two successive vectors. 



Geometric addition agrees, therefore, with 

 algebraic addition in being commutative. Fig. 14. 



49. The vector, as the geometric symbol of a translation, has 

 length, direction, and sense ; but it is not restricted to any 

 definite position, the same translation being represented by all 

 equal and parallel vectors. We express this by saying that two 

 vectors are equal if they are of the same length, direction, and sense. 



Translations are not the only magnitudes in mechanics 

 that can be represented by vectors. We shall see later that 

 velocities, accelerations, moments of couples, etc., can all be 

 represented by vectors and are therefore compounded into 

 resultants and resolved into components by geometric addition 

 and subtraction. In this lies the importance of this subject 

 which in its special application to translations might appear too 

 simple and self-evident to require extended presentation. 



The case when the vectors represent concurrent forces is 

 probably known to the student from elementary physics as the 

 "parallelogram " or " polygon " of forces. 



50. A translation may be resolved into two or more translations 

 by resolving its vector into components. 





