48 TREATISE ON ALTERNATING CURRENTS. 



MULTIPLICATION OF VECTORS. 



37. Vector Products. Consider any two vectors OP 

 and OQ (Fig. 14), and let OP = ON + NP, where NP is at right 

 angles to OQ. 



There are two products to 

 take into consideration, viz. the 

 product 



OQ. ON 

 and the product 



OQ .NP 



FIG. 14. 



To interpret these, suppose 



that the vector OP represents a displacement produced by the 

 action of a force which is represented by the vector OQ. 

 The product 



OQ . ON 



then represents the work done by the force in moving its point 

 of application from the point to the point N. This product is 

 essentially scalar, since it is the increase of energy of the system 

 on which the force Q acts. 

 The product 



OQ .NP 



represents the moment of the force about an axis through the 

 point P at right angles to both OP and NP, and is a vector 

 at right angles to the plane OPN. 



It is the scalar product of two vectors with which we are 

 chiefly interested, so we leave the vector product for the present. 



By reference to Fig. 14, we see that 



OQ .ON = OQ . OP cos 



where is the angle between the directions of OQ and OP respec- 

 tively; hence the scalar product of two vectors is 

 the product of their lengths multiplied by the 

 cosine of the angle between their respective 

 directions. 



Let the vector OP be represented algebraically by 



a 4- M> 

 and the vector OQ by 



a 1 + kV 



