10 NUMBER. [INT. I. 



the diagonal of the parallelogram formed from the two given 

 vectors as sides, or as given by applying the initial point of the 

 second vector to the terminal point of the first, and constructing a 

 new vector joining the initial point of the first to the terminal 

 point of the second. Either of these geometrical processes shows 

 that the addition of vectors is commutative. The addition of 

 vectors in this manner is known as geometrical addition. When 

 such geometrical addition of vectors is meant, as distinguished 

 from arithmetical addition of their tensors, we shall denote the 

 quantities to be considered as vectors by placing a bar over the 

 quantity otherwise used for the tensor ; e.g. the equations 



Yj + Zk, 



R = R = 

 are examples of vector and scalar equations respectively. 



7. Geometric Multiplication. As we have seen in 4, 

 by direct multiplication, the product of the two vectors 



.#! = Xji + Yjj + ZJs and E a = X t i+ YJ + ZJc 

 is - (X,X, + Y,Y 2 + Z&) + ( Y& - Z, F 2 ) i 



+ (Z,X 2 - XA)j + (X,Y, - FA) k. 

 Of this the scalar part 



has an important geometrical meaning. The direction cosines of 

 the vector R being denoted by cos (Rx), cos (Ry\ cos (Rz), we have 

 by the definition of the projections of R, 



Consequently, 



cos & cos 

 + cos (Riy) cos (R$) + cos (R^z) cos (R^z)}. 



The factor in the brackets is equal to the cosine of the angle 

 between the directions of R l and J? 2 . Consequently the scalar 

 part of the product of the vectors R^ and R. 2 , or the scalar product 

 of the two vectors, which will be denoted by the notation SR^ is 

 equal to minus the product of the tensor of either multiplied by 

 the projection of the other on its own direction 



cos 



