By Tuckerman M.
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Yn . 32) was obtained by considering the axiom (b) of the affine spaces and the fact that PO ¼ ÀOP. The conclusion is that the affine coordinates of the vector PQ can be written using only the affine coordinates of the points P and Q. 2 Distance in Affine Spaces Consider the case of an affine space which is defined by using a linear space on which a scalar product has been introduced. 6 Spaces 27 dðA; BÞ ¼ ðAB Á ABÞ1=2 ð2:33Þ By choosing an affine coordinate system consisting of a point O and an orthonormal basis, one obtains an image of the affine space similar to that of the space in Euclidean geometry.
The coordinates of OP in the basis e1 ; e2 ; . ; en are denoted ðx1 ; x2 ; . ; xn Þ. In this case, the following relation can be written: OP ¼ x1 e1 þ x2 e2 þ Á Á Á þ xn en xi ei ð2:31Þ The coefficients ðx1 ; x2 ; . ; xn Þ represent the affine coordinates of the point P. Note that these coordinates depend on the basis e1 ; e2 ; . ; en of the linear space V and the origin O (element of M). Assume the point Q 2 M, of affine coordinates y1 ; y2 ; . ; yn . 32) was obtained by considering the axiom (b) of the affine spaces and the fact that PO ¼ ÀOP.
Similar sentences can be formulated in the case of pseudovectors. 2 2 Algebraic Structures. Spaces. Reference Frames Operation of Addition The addition makes sense only between quantities with the same physical dimension. The addition of scalars (pseudoscalars) can be done in a given reference frame. It consists in the sum of the numerical values of the scalars (pseudoscalars). Assume two vectors (pseudovectors) u and v, of components uk and vk ðk ¼ 1; 2; 3Þ in a given reference frame. The sum of the two vectors (pseudovectors) is a vector (pseudovector), denoted w, whose components are equal to the sum of the numerical values of the components of the two vectors.
