GRAPHICAL ALGEBRA. 



79 



The fact that the vector F is the vector-sum according to the parallelogram-law 

 of the two vectors A and B will be denoted by the vector-equation. 



(a) 



A+B 



This equation can be considered as equivalent to the three scalar equations 



(a') 



A x +B x 



F y = A y +B, 



F 2 = A t +B, 



which express the projections of F as the scalar sum of the projections of A and of 

 B (fig. 71). The scalar-sum of the tensors A+B must be carefully distinguished 

 from the scalar value or tensor |A + B|of the vector-sum. There will be identity 

 between the scalar sum of the tensors and the tensor of vector-sum when the two 

 given vectors have the same direction, and between the scalar differences of the 

 tensors and the tensor of the vector-sum when the two given vectors have opposite 

 directions. 



A scalar quantity which is equal to the product of the tensors of two given 

 vectors and the cosine of the included angle will be called the scalar product of the 

 two given vectors. When the given vectors are 

 A and B, their scalar product shall be denoted 

 by A.B, thus 



(b) A.B = AB cos 6 



By the fundamental formula? of analytical geom- 

 etry it is easily verified that the scalar product 

 is equal to the sum of the products of the rec- 

 tangular components of the given vectors, 



K 



-Vector-addition. 



(jb') A.B = A x B x +A y B y +A : B z 



The vector-operations defined by the pre- 

 ceding formulae are symmetrical with respect to 

 the two given vectors A and B. In the vector- 

 formula? the symbols for the vectors can there- Fig. 71 

 fore be commutated 

 (c) A+B = B+A A.B = B.A 



We shall define finally an important unsymmetric vector-operation, in which 

 this commutation of the symbols will no more be allowed. The succession of the 

 symbols will be used to serve an important purpose, namely, to distinguish between 

 opposite directions in space. In order to give the definition of this operation, we must 

 first make an important remark concerning the geometry of translations and rotations. 



Let an axis in space be given. Two opposite translations will be possible along 

 it, and two opposite rotations will be possible around it. We must agree upon a 

 definite connection by which we can define the positive direction of rotation as soon 

 as the positive direction of translation is chosen, and vice versa. We shall attain 

 this by the rule of the positive or right-handed screw. When this screw moves in its 



