20 GEOMETRY OF VECTORS. [CHAP. IL. 



The moment about a line of a vector localised at a point is 

 defined to be the moment about the line of a vector with the 

 same magnitude, direction, and sense, but localised in a line 

 through the point. 



17. Moments about a point. When we are concerned 

 with vectors localised in lines that lie in a plane, or with vectors 

 localised at points in a plane and having their directions parallel 

 to that plane, the moments of these vectors about a line perpen- 

 dicular to the plane are defined to be the moments of the vectors 

 about the point where this line meets the plane. For the sake of 

 simplicity we may also give the following definition : 



The moment about a point of a vector in a plane through 

 and localised in a line L is the product, with a certain sign, of the 

 magnitude of the vector and the perpendicular upon L from the 

 point 0. The rule of signs is that when the sense of the vector is- 

 opposite to that of the motion of the hands of a watch, placed 

 with its face in the plane, and the centre of its face at 0, the sign 

 is +, otherwise the sign is . The watch must be supposed to 

 have its face always turned the same way. 



18. Lemma. The moment about a point of a vector 

 localised at a point A is identical with the moment about of the 

 resolved part of the vector at right angles to OA. 



This follows at once by using the first form of the definition, 



and it can be deduced 

 from the second form as 

 follows : 



Let be the angle 

 the direction of the vector 

 makes with the line AO, 

 and draw ON at right 

 angles to the line of the 

 vector. The magnitude of 

 the resolved part of the 

 vector at right angles to 

 AO is E sin 0, where E is 

 the magnitude of the vec- 

 tor. The perpendicular from on the line of the vector is the 

 line ON, and it is equal to OA . sin 0. 



