28 GEOMETRY OF VECTORS. [CHAP. II. 



plane, or to a couple whose axis is perpendicular to the plane, or 

 to zero. 



The single vector, and the couple, in the cases where the 

 system is equivalent to a single vector or a couple, are deter 

 minate and unique. 



The conditions of equivalence of two systems of vectors localised 

 in lines lying in a plane are that when one system is equivalent 

 to a single vector the other is equivalent to a single vector of the 

 same magnitude and sense localised in the same line, when one 

 system is equivalent to a couple the other is equivalent to a 

 couple of the same magnitude and sense, when one system is 

 equivalent to zero the other is equivalent to zero. 



26. Analysis of vectors localised in lines in a plane. 



Take any origin 0, and any rectangular axes of as, y in the plane. 



Let X lt Y l be the resolved parts 

 parallel to the axes of one of 

 the vectors, and let P a (x l} y^) be 

 any point on the line in which it 

 x * is localised. Then the moment 



of the couple, when this vector is 

 replaced by an equal vector local 

 ised in a line through and a 

 couple, is the moment of this 

 _ x vector about 0, i.e. it is the sum 



of the moments of the resolved 

 Fig. 25. 



parts X lt Y 1 about 0, and this 



sum is a! 1 Y l yiX l . Hence the whole system is equivalent to a 

 vector localised in a line through 0, whose resolved parts parallel 

 to the axes are %X and 2F, and a couple in the plane, whose 

 moment is !LxY 



27. General analysis of vectors localised in lines. 



Take any origin 0, and any rectangular axes of x, y, z. Let 

 X, F, Z be the resolved parts parallel to the axes of one of the 

 vectors, and x, y, z the coordinates of a point in the line in which 

 it is localised. Introduce a pair of equal and opposite vectors 

 localised in a line through parallel to the line of this vector, 

 and resolve them into components localised in the axes. The 



