4, 5] PROJECTION. GEOMETRIC PRODUCT. MOMENTS. 7 



If cos A, cos \i y cos v are the direction cosines of a second vector s~ 9 



cos I = 9 



2W "2z 



= 7 COS V = 7 



multiplying by s 2 we have the expression symmetrical with respect 

 to both vectors 



10) S t 



This expression, which may be denned either as the product of the 

 tensors of the vectors and the cosine of their included angle , or as 

 the tensor of either multiplied by the value of the projection on its 

 direction of the other, is so important that it has received a special 

 name, and will be called the geometric product of the two vectors. 

 It is not a vector, but is essentially a scalar quantity, and its negative 

 was called by Hamilton the scalar product of the vectors. 



The condition of perpendicularity of two vectors is that their 

 geometric product vanishes. 



2 = 0. 



5. Moments. Consider a fixed vector AB, Fig. 2. The product 

 of the length AB and the perpendicular distance of from AB is 

 called the moment of 

 AB about 0. It is 

 arithmetically equal 

 to twice the area of 

 the triangle OAB. 

 The sign of the 

 moment will change 

 with the direction of 

 AB. If we draw 

 a line through 

 whose length is 

 equal to the magni- 

 tude of the moment 

 and whose direction 

 is perpendicular to 

 the plane A B, this 

 line is called the axis 



of the moment, and in a certain way represents the latter. We shall 

 draw it in such a direction that a person standing on with his back 

 against the axis would see motion from A to B as from right 

 to left. 



