GRAPHICAL ALGEBRA. 8 1 



The relation of the vector-product F to the vector-factors A and B is illustrated 

 by fig. 73 : F is directed along the normal to the plane which contains A and B; the 

 positive rotation around F transfers the first vector-factor A into the second B ; 

 and F has the scalar value F, which is given by the formula 



{d') F = AB sin d 



or which is represented geometrically by the area of the parallelogram which has 

 sides representing the vector-factors. 



It follows immediately from the definitions that when we commutate the vec- 

 tors A and B, we get the vector F, which is directed oppositely to F, thus 



0) BXA = -AXB 



When we bring coordinates into application we shall agree to use consistently 

 what we shall call a positive system of coordinates. Let the positive direction along 

 each of the rectangular axes be chosen. The corresponding positive rotation around 

 an axis will then either be a rotation in or against that direction which is defined by 

 the succession of letters X, Y, Z, X, . . . In the first case the system will be 

 called positive, in the second case negative. Thus when the system is positive, the 

 positive rotation around Z will go from X to Y, the positive rotation around X will go 

 from Y to Z, the positive rotation around Y will go from Z to X. A positive system 

 of coordinates, of which we shall make a frequent use, is one which has its axis of X 

 directed toward the east, its axis of Y directed toward the north, and its axis of Z 

 directed upward. (See fig. 74.) 



When we use a positive system of coordinates, it is easily verified by the funda- 

 mental formulae of analytical geometry that the rectangular components F x , F y , Fz of 

 the vector-product F are 



(/) F x = A y B, - A z B y F y = A Z B X - A X B Z F z = A x B y -A y B x 



The vector-equation (d) may be considered as a shortened symbolic expression for 

 the three equations (/). Equations (/) also at once lead to the result expressed by 

 equation (e), that the vector-product changes its sign when the succession of the 

 vector- factors is interchanged ; for we get F x , F y , and F z when in equations (/) 

 we change A x with B x , A y with B y , and A z with B z . 



154. Consistent Use of Rectangular Components in Graphical Vector- Algebra. 



As drawings are two-dimensional, our methods can deal directly only with two- 

 dimensional fields. Vector-fields in space must be treated indirectly. We have 

 introduced for this the method of solving the three-dimensional field into fields tan- 

 gential to and normal to a set of surfaces (section 118). The normal field may be 

 treated as a two-dimensional scalar, while the tangential field represents a true two- 

 dimensional vector. Our subject will therefore be that of developing graphical 

 methods for performing mathematical operations upon these two-dimensional vectors. 

 One general method presents itself at once. We can introduce two sets of curves 

 cutting each other under right angles, and use them as coordinate-curves. In the 

 simplest case the two sets of curves will be two sets of parallel lines, which are mutually 



