. Carnegie Institution of Washington publication. 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') Ax+Bx Fy = Ay+B, F2 = At+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 s
. Carnegie Institution of Washington publication. 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') Ax+Bx Fy = Ay+B, F2 = At+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 , thus (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') = AxBx+AyBy+A:Bz 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 = 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 rota
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