Projective Vector Algebra An Algebra of Vectors Independent of the Axioms of Co
Projective Vector Algebra An Algebra of Vectors Independent of the Axioms of Co
Ludwik Silberstein
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If the six points X, etc. , are fixed (prescribed), so are also, of course, the crosses P lt P z, P 3 . The choice of the T-plane influences only the values of the numbers x, ... Z' (after the standard vectors a, a' have been fixed by convention). Now, whatever the straight PjPgPg, we can, if we so desire, lay our T-plane through it. Then we shall have the remarkably simple relations between the six numbers : xx' = yy'=zz'. This, however, by the way only. The purpose of the present section has ...been to show that the vector algebra in question gives a simple and perfectly symmetrical proof of Pascal's theorem. Hilbert's proof, which may be looked up in his quoted book, is considerably more intricate. Schur's proof, of course, is much more elegant than Hilbert's, but is based upon the postulates of congruence or motion, which according to our plan were to be strictly avoided. 15. Vector sum for variable /-line. Conic ranges and pencils. Let us consider two fixed coinitial, non-collinear vectors JL = OX and Y = Y.
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