"En las matem\u00E1ticas, \u00E1lgebra geom\u00E9trica es un t\u00E9rmino aplicado a la teor\u00EDa de las \u00E1lgebras de Clifford y teor\u00EDas relacionadas, siguiendo un libro del mismo t\u00EDtulo por Emil Artin. Este t\u00E9rmino tambi\u00E9n ha tenido reciente uso en los tratamientos de la misma \u00E1rea en la literatura f\u00EDsica. En David Hestenes et al. \u00E1lgebra geom\u00E9trica es una reinterpretaci\u00F3n de las \u00E1lgebras de Clifford sobre los reales (lo que se afirma como una vuelta al nombre y a la interpretaci\u00F3n originales previstos por William Clifford). Los n\u00FAmeros reales se utilizan como escalares en un espacio vectorial V. Desde ahora en adelante, un vector es algo en V mismo. El (producto exterior, o ) \u2227 se define tal que se genere el \u00E1lgebra graduada (\u00E1lgebra exterior de Hermann Grassmann) de \u039Bn Vn de multivectores. El \u00E1lgebra geom\u00E9tric"@es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "\uAE30\uD558\uC801 \uB300\uC218\uD559"@ko . . . . . . . . . . . "Alg\u00E8bre g\u00E9om\u00E9trique (structure)"@fr . . . . . "220"^^ . . . . . . . "Reversed orientation corresponds to negating the exterior product."@en . "\u0397 \u03B3\u03B5\u03C9\u03BC\u03B5\u03C4\u03C1\u03B9\u03BA\u03AE \u03AC\u03BB\u03B3\u03B5\u03B2\u03C1\u03B1 (\u0393.\u0391) \u03B5\u03AF\u03BD\u03B1\u03B9 \u03BC\u03B9\u03B1 \u03C4\u03BF\u03C5 \u03B4\u03B9\u03B1\u03BD\u03C5\u03C3\u03BC\u03B1\u03C4\u03B9\u03BA\u03BF\u03CD \u03C7\u03CE\u03C1\u03BF\u03C5 \u03C0\u03AC\u03BD\u03C9 \u03B1\u03C0\u03CC \u03C4\u03BF \u03C0\u03B5\u03B4\u03AF\u03BF \u03C4\u03C9\u03BD \u03C0\u03C1\u03B1\u03B3\u03BC\u03B1\u03C4\u03B9\u03BA\u03CE\u03BD \u03B1\u03C1\u03B9\u03B8\u03BC\u03CE\u03BD \u03C0\u03C1\u03BF\u03B9\u03BA\u03B9\u03C3\u03BC\u03AD\u03BD\u03BF \u03BC\u03B5 \u03BC\u03B9\u03B1 . \u039F \u03CC\u03C1\u03BF\u03C2 \u03B5\u03C0\u03AF\u03C3\u03B7\u03C2 \u03BC\u03B5\u03C1\u03B9\u03BA\u03AD\u03C2 \u03C6\u03BF\u03C1\u03AD\u03C2 \u03C7\u03C1\u03B7\u03C3\u03B9\u03BC\u03BF\u03C0\u03BF\u03B9\u03B5\u03AF\u03C4\u03B1\u03B9 \u03C9\u03C2 \u03C3\u03C5\u03BB\u03BB\u03BF\u03B3\u03B9\u03BA\u03CC\u03C2 \u03CC\u03C1\u03BF\u03C2 \u03B3\u03B9\u03B1 \u03C4\u03B7\u03BD \u03C0\u03C1\u03BF\u03C3\u03AD\u03B3\u03B3\u03B9\u03C3\u03B7 \u03C3\u03C4\u03B7 \u03BA\u03BB\u03B1\u03C3\u03C3\u03B9\u03BA\u03AE, \u03C5\u03C0\u03BF\u03BB\u03BF\u03B3\u03B9\u03C3\u03C4\u03B9\u03BA\u03AE \u03BA\u03B1\u03B9 \u03C3\u03C7\u03B5\u03C4\u03B9\u03BA\u03B9\u03C3\u03C4\u03B9\u03BA\u03AE \u03B3\u03B5\u03C9\u03BC\u03B5\u03C4\u03C1\u03AF\u03B1 \u03C0\u03BF\u03C5 \u03B5\u03C6\u03B1\u03C1\u03BC\u03CC\u03B6\u03B5\u03B9 \u03B1\u03C5\u03C4\u03AD\u03C2 \u03C4\u03B9\u03C2 \u03AC\u03BB\u03B3\u03B5\u03B2\u03C1\u03B5\u03C2. \u039F \u03C0\u03BF\u03BB\u03BB\u03B1\u03C0\u03BB\u03B1\u03C3\u03B9\u03B1\u03C3\u03BC\u03CC\u03C2 \u039A\u03BB\u03AF\u03C6\u03BF\u03C1\u03BD\u03C4 \u03C0\u03BF\u03C5 \u03BF\u03C1\u03AF\u03B6\u03B5\u03B9 \u03C4\u03BF \u0393.\u03A3 \u03C9\u03C2 \u03BC\u03BF\u03BD\u03AC\u03B4\u03B1 \u03B4\u03B1\u03BA\u03C4\u03C5\u03BB\u03AF\u03BF\u03C5 \u03BF\u03BD\u03BF\u03BC\u03AC\u03B6\u03B5\u03C4\u03B1\u03B9 \u03B3\u03B5\u03C9\u03BC\u03B5\u03C4\u03C1\u03B9\u03BA\u03CC \u03C0\u03C1\u03BF\u03CA\u03CC\u03BD. \u0397 \u03BB\u03B5\u03B9\u03C4\u03BF\u03C5\u03C1\u03B3\u03AF\u03B1 \u03B5\u03BA\u03C4\u03CC\u03C2 \u03CC\u03C4\u03B9 \u03C3\u03C5\u03BD\u03B4\u03C5\u03AC\u03B6\u03B5\u03B9 \u03B1\u03C5\u03C4\u03AE\u03BD \u03C3\u03B5 \u03B3\u03B5\u03BD\u03B9\u03BA\u03AD\u03C2 \u03B3\u03C1\u03B1\u03BC\u03BC\u03AD\u03C2 \u03BC\u03B5 \u03C0\u03BF\u03BB\u03C5\u03B4\u03B9\u03B1\u03BD\u03CD\u03C3\u03BC\u03B1\u03C4\u03B1 , \u03C4\u03B1 \u03BF\u03C0\u03BF\u03AF\u03B1 \u03B5\u03AF\u03BD\u03B1\u03B9 \u03C4\u03B1 \u03C3\u03C4\u03BF\u03B9\u03C7\u03B5\u03AF\u03B1 \u03C4\u03BF\u03C5 \u03B4\u03B1\u03BA\u03C4\u03C5\u03BB\u03AF\u03BF\u03C5. \u0391\u03C5\u03C4\u03CC \u03C0\u03B5\u03C1\u03B9\u03BB\u03B1\u03BC\u03B2\u03AC\u03BD\u03B5\u03B9, \u03BC\u03B5\u03C4\u03B1\u03BE\u03CD \u03AC\u03BB\u03BB\u03C9\u03BD \u03B4\u03C5\u03BD\u03B1\u03C4\u03BF\u03C4\u03AE\u03C4\u03C9\u03BD, \u03AD\u03BD\u03B1 \u03BA\u03B1\u03BB\u03AC \u03BA\u03B1\u03B8\u03BF\u03C1\u03B9\u03C3\u03BC\u03AD\u03BD\u03BF \u03C4\u03C5\u03C0\u03B9\u03BA\u03CC \u03AC\u03B8\u03C1\u03BF\u03B9\u03C3\u03BC\u03B1 \u03B2\u03B1\u03B8\u03BC\u03C9\u03C4\u03CC \u03BA\u03B1\u03B9 \u03B4\u03B9\u03B1\u03BD\u03C5\u03C3\u03BC\u03B1\u03C4\u03B9\u03BA\u03CC."@el . . . "Une alg\u00E8bre g\u00E9om\u00E9trique est, en math\u00E9matiques, une structure alg\u00E9brique, similaire \u00E0 une alg\u00E8bre de Clifford r\u00E9elle, mais dot\u00E9e d'une interpr\u00E9tation g\u00E9om\u00E9trique mise au point par David Hestenes, reprenant les travaux de Hermann Grassmann et William Kingdon Clifford (le terme est aussi utilis\u00E9 dans un sens plus g\u00E9n\u00E9ral pour d\u00E9crire l'\u00E9tude et l'application de ces alg\u00E8bres : l'alg\u00E8bre g\u00E9om\u00E9trique est l'\u00E9tude des alg\u00E8bres g\u00E9om\u00E9triques). Le but avou\u00E9 de ce physicien th\u00E9oricien et p\u00E9dagogue est de fonder un langage propre \u00E0 unifier les manipulations symboliques en physique, dont les nombreuses branches pratiquent aujourd'hui, pour des raisons historiques, des formalismes diff\u00E9rents (tenseurs, matrices, torseurs, analyse vectorielle, utilisation de nombres complexes, spineurs, quaternions, formes diff\u00E9rentielles\u2026). Le nom choisi par David Hestenes (geometric algebra) est celui que Clifford voulait donner \u00E0 son alg\u00E8bre. L'alg\u00E8bre g\u00E9om\u00E9trique se veut utile dans les probl\u00E8mes de physique qui impliquent des rotations, des phases ou des nombres imaginaires. Ses partisans disent qu'elle fournit une description plus compacte et intuitive de la m\u00E9canique quantique et classique, de la th\u00E9orie \u00E9lectromagn\u00E9tique et de la relativit\u00E9. Les applications actuelles de l'alg\u00E8bre g\u00E9om\u00E9trique incluent la vision par ordinateur, la biom\u00E9canique ainsi que la robotique et la dynamique des vols spatiaux."@fr . . . . . . . . . . . . . "\u00C0lgebra geom\u00E8trica"@ca . . . . . "\uAE30\uD558\uC801 \uB300\uC218\uD559(\uC601\uC5B4: Geometric Algebra (GA))\uC740 \uC218\uD559\uC5D0\uC11C \uD074\uB9AC\uD37C\uB4DC \uB300\uC218\uC758 \uAE30\uD558\uD559\uC801 \uD574\uC11D\uC774\uBA70 3\uCC28\uC6D0 \uACF5\uAC04\uC5D0\uC11C \uC9C1\uC811\uC801\uC73C\uB85C \uACF5\uAC04\uACFC \uC2DC\uAC04\uC744 \uBCA1\uD130 \uBBF8\uC801\uBD84\uBCF4\uB2E4 \uAC04\uB2E8\uD558\uAC8C \uD45C\uD604\uD558\uACE0 \uD574\uC11D\uD560 \uC218 \uC788\uB2E4. \uAE30\uD558\uC801 \uB300\uC218\uD559\uC740 \uC218\uD559\uC801 \uBB38\uC81C\uC5D0\uC11C \uD68C\uC804, \uC704\uC0C1\uC774\uB098, \uBCF5\uC18C\uC218\uB97C \uC0AC\uC6A9\uD560 \uACBD\uC6B0 \uBB38\uC81C\uB97C \uAC04\uB2E8\uD558\uACE0 \uC54C\uAE30 \uC27D\uAC8C \uD45C\uD604\uD560 \uC218 \uC788\uAE30 \uB54C\uBB38\uC5D0 \uBB3C\uB9AC\uC758 \uACE0\uC804\uC5ED\uD559, \uC591\uC790\uC5ED\uD559, \uC804\uC790\uAE30\uD559, \uB85C\uBD07\uACF5\uD559, \uCEF4\uD4E8\uD130 \uBE44\uC804\uACFC \uCEF4\uD4E8\uD130 \uADF8\uB798\uD53D \uB4F1\uC5D0 \uC751\uC6A9\uB418\uACE0\uC788\uB2E4."@ko . . . . . . . . . . . . . . . . . "Geometric interpretation of grade- elements in a real exterior algebra for , , , . The exterior product of vectors can be visualized as any -dimensional shape ; with magnitude , and orientation defined by that on its -dimensional boundary and on which side the interior is."@en . . . . . . . . "\uAE30\uD558\uC801 \uB300\uC218\uD559(\uC601\uC5B4: Geometric Algebra (GA))\uC740 \uC218\uD559\uC5D0\uC11C \uD074\uB9AC\uD37C\uB4DC \uB300\uC218\uC758 \uAE30\uD558\uD559\uC801 \uD574\uC11D\uC774\uBA70 3\uCC28\uC6D0 \uACF5\uAC04\uC5D0\uC11C \uC9C1\uC811\uC801\uC73C\uB85C \uACF5\uAC04\uACFC \uC2DC\uAC04\uC744 \uBCA1\uD130 \uBBF8\uC801\uBD84\uBCF4\uB2E4 \uAC04\uB2E8\uD558\uAC8C \uD45C\uD604\uD558\uACE0 \uD574\uC11D\uD560 \uC218 \uC788\uB2E4. \uAE30\uD558\uC801 \uB300\uC218\uD559\uC740 \uC218\uD559\uC801 \uBB38\uC81C\uC5D0\uC11C \uD68C\uC804, \uC704\uC0C1\uC774\uB098, \uBCF5\uC18C\uC218\uB97C \uC0AC\uC6A9\uD560 \uACBD\uC6B0 \uBB38\uC81C\uB97C \uAC04\uB2E8\uD558\uACE0 \uC54C\uAE30 \uC27D\uAC8C \uD45C\uD604\uD560 \uC218 \uC788\uAE30 \uB54C\uBB38\uC5D0 \uBB3C\uB9AC\uC758 \uACE0\uC804\uC5ED\uD559, \uC591\uC790\uC5ED\uD559, \uC804\uC790\uAE30\uD559, \uB85C\uBD07\uACF5\uD559, \uCEF4\uD4E8\uD130 \uBE44\uC804\uACFC \uCEF4\uD4E8\uD130 \uADF8\uB798\uD53D \uB4F1\uC5D0 \uC751\uC6A9\uB418\uACE0\uC788\uB2E4."@ko . . . . . . . . . "\u0393\u03B5\u03C9\u03BC\u03B5\u03C4\u03C1\u03B9\u03BA\u03AE \u03AC\u03BB\u03B3\u03B5\u03B2\u03C1\u03B1"@el . . . . . . "Orientation defined by an ordered set of vectors."@en . . . . "1124849689"^^ . "En las matem\u00E1ticas, \u00E1lgebra geom\u00E9trica es un t\u00E9rmino aplicado a la teor\u00EDa de las \u00E1lgebras de Clifford y teor\u00EDas relacionadas, siguiendo un libro del mismo t\u00EDtulo por Emil Artin. Este t\u00E9rmino tambi\u00E9n ha tenido reciente uso en los tratamientos de la misma \u00E1rea en la literatura f\u00EDsica. En David Hestenes et al. \u00E1lgebra geom\u00E9trica es una reinterpretaci\u00F3n de las \u00E1lgebras de Clifford sobre los reales (lo que se afirma como una vuelta al nombre y a la interpretaci\u00F3n originales previstos por William Clifford). Los n\u00FAmeros reales se utilizan como escalares en un espacio vectorial V. Desde ahora en adelante, un vector es algo en V mismo. El (producto exterior, o ) \u2227 se define tal que se genere el \u00E1lgebra graduada (\u00E1lgebra exterior de Hermann Grassmann) de \u039Bn Vn de multivectores. El \u00E1lgebra geom\u00E9trica es el \u00E1lgebra generada por el producto geom\u00E9trico (el cual es pensado como fundamental) con (para todos los A, B, C) 1. \n* Asociatividad 2. \n* Distributividad sobre la adici\u00F3n de multivectores: A(B + C) = A B + A C y (A + B)C = A C + B C 3. \n* La para cualquier \"vector\" (un elemento de grado uno) a, a\u00B2 es un escalar (n\u00FAmero real) Llamamos esta \u00E1lgebra un \u00E1lgebra geom\u00E9trica . El punto distintivo de esta formulaci\u00F3n es la correspondencia natural entre las entidades geom\u00E9tricas y los elementos del \u00E1lgebra asociativa. La conexi\u00F3n entre las \u00E1lgebra de Clifford y las formas cuadr\u00E1ticas vienen de la propiedad de contracci\u00F3n. Esta regla tambi\u00E9n da al espacio una m\u00E9trica definida por el naturalmente derivado producto interno. Debe ser observado que en \u00E1lgebra geom\u00E9trica en toda su generalidad no hay restricci\u00F3n ninguna en el valor del escalar, puede suceder que sea negativa, incluso cero (en tal caso, la posibilidad de un producto interno est\u00E1 eliminada si se requiere ). El producto escalar usual y el producto cruzado tradicional del \u00E1lgebra vectorial (en ) hallan sus lugares en el \u00E1lgebra geom\u00E9trica como el producto interno: (que es sim\u00E9trico) y el producto externo: con: (que es antisim\u00E9trico). Relevante es la distinci\u00F3n entre los vectores axiales y polares en el \u00E1lgebra vectorial, que es natural en \u00E1lgebra geom\u00E9trica como la mera distinci\u00F3n entre los vectores y los bivectores (elementos de grado dos). El i aqu\u00ED es la unidad del 3-espacio euclidiano, lo que establece una dualidad entre los vectores y los bivectores, y se lo llama as\u00ED debido a la propiedad prevista i\u00B2 = -1. Un ejemplo \u00FAtil es , y generar , un caso del \u00E1lgebra geom\u00E9trica llamada \u00E1lgebra del espacio-tiempo por Hestenes. El tensor del campo electromagn\u00E9tico, en este contexto, se convierte en simplemente un bivector donde la unidad imaginaria es el elemento de volumen, dando un ejemplo de la reinterpretaci\u00F3n geom\u00E9trica de los \"trucos tradicionales\". en esta m\u00E9trica de Lorentz tienen la misma expresi\u00F3n que la rotaci\u00F3n en el espacio euclidiano, donde es, por supuesto, el bivector generado por el tiempo y las direcciones del espacio implicadas, mientras que en el caso euclidiano es el bivector generado por las dos direcciones del espacio, consolidando la \"analog\u00EDa\" casi hasta la identidad."@es . "N vector positive.svg"@en . . . . "Une alg\u00E8bre g\u00E9om\u00E9trique est, en math\u00E9matiques, une structure alg\u00E9brique, similaire \u00E0 une alg\u00E8bre de Clifford r\u00E9elle, mais dot\u00E9e d'une interpr\u00E9tation g\u00E9om\u00E9trique mise au point par David Hestenes, reprenant les travaux de Hermann Grassmann et William Kingdon Clifford (le terme est aussi utilis\u00E9 dans un sens plus g\u00E9n\u00E9ral pour d\u00E9crire l'\u00E9tude et l'application de ces alg\u00E8bres : l'alg\u00E8bre g\u00E9om\u00E9trique est l'\u00E9tude des alg\u00E8bres g\u00E9om\u00E9triques). Le but avou\u00E9 de ce physicien th\u00E9oricien et p\u00E9dagogue est de fonder un langage propre \u00E0 unifier les manipulations symboliques en physique, dont les nombreuses branches pratiquent aujourd'hui, pour des raisons historiques, des formalismes diff\u00E9rents (tenseurs, matrices, torseurs, analyse vectorielle, utilisation de nombres complexes, spineurs, quaternions, forme"@fr . . . "En matem\u00E0tiques, \u00E0lgebra geom\u00E8trica \u00E9s un terme aplicat a la teoria de les \u00E0lgebres de Clifford i teories relacionades, seguint un llibre del mateix t\u00EDtol d'Emil Artin. Aquest terme tamb\u00E9 ha tingut recent \u00FAs en els tractaments de la mateixa \u00E0rea en la literatura de f\u00EDsica. El producte escalar usual i el producte creuat tradicional de l'\u00E0lgebra vectorial (a ) troben els seus llocs en l'\u00E0lgebra geom\u00E8trica com el producte intern: (que \u00E9s sim\u00E8tric) I el producte extern: con: (que \u00E9s antisim\u00E8tric)."@ca . . . . . . . . . . . . . . . "82293"^^ . . . . . . . . . . . . . "\u0397 \u03B3\u03B5\u03C9\u03BC\u03B5\u03C4\u03C1\u03B9\u03BA\u03AE \u03AC\u03BB\u03B3\u03B5\u03B2\u03C1\u03B1 (\u0393.\u0391) \u03B5\u03AF\u03BD\u03B1\u03B9 \u03BC\u03B9\u03B1 \u03C4\u03BF\u03C5 \u03B4\u03B9\u03B1\u03BD\u03C5\u03C3\u03BC\u03B1\u03C4\u03B9\u03BA\u03BF\u03CD \u03C7\u03CE\u03C1\u03BF\u03C5 \u03C0\u03AC\u03BD\u03C9 \u03B1\u03C0\u03CC \u03C4\u03BF \u03C0\u03B5\u03B4\u03AF\u03BF \u03C4\u03C9\u03BD \u03C0\u03C1\u03B1\u03B3\u03BC\u03B1\u03C4\u03B9\u03BA\u03CE\u03BD \u03B1\u03C1\u03B9\u03B8\u03BC\u03CE\u03BD \u03C0\u03C1\u03BF\u03B9\u03BA\u03B9\u03C3\u03BC\u03AD\u03BD\u03BF \u03BC\u03B5 \u03BC\u03B9\u03B1 . \u039F \u03CC\u03C1\u03BF\u03C2 \u03B5\u03C0\u03AF\u03C3\u03B7\u03C2 \u03BC\u03B5\u03C1\u03B9\u03BA\u03AD\u03C2 \u03C6\u03BF\u03C1\u03AD\u03C2 \u03C7\u03C1\u03B7\u03C3\u03B9\u03BC\u03BF\u03C0\u03BF\u03B9\u03B5\u03AF\u03C4\u03B1\u03B9 \u03C9\u03C2 \u03C3\u03C5\u03BB\u03BB\u03BF\u03B3\u03B9\u03BA\u03CC\u03C2 \u03CC\u03C1\u03BF\u03C2 \u03B3\u03B9\u03B1 \u03C4\u03B7\u03BD 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"\u00C1lgebra geom\u00E9trica"@es . . . "In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called multivectors. Compared to other formalisms for manipulating geometric objects, geometric algebra is noteworthy for supporting vector division and addition of objects of different dimensions. The geometric product was first briefly mentioned by Hermann Grassmann, who was chiefly interested in developing the closely related exterior algebra. In 1878, William Kingdon Clifford greatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his honor (although Clifford himself chose to call them \"geometric algebras\"). Clifford defined the Clifford algebra and its product as a unification of the Grassmann algebra and Hamilton's quaternion algebra. Adding the dual of the Grassmann exterior product (the \"meet\") allows the use of the Grassmann\u2013Cayley algebra, and a conformal version of the latter together with a conformal Clifford algebra yields a conformal geometric algebra providing a framework for classical geometries. In practice, these and several derived operations allow a correspondence of elements, subspaces and operations of the algebra with geometric interpretations. For several decades, geometric algebras went somewhat ignored, greatly eclipsed by the vector calculus then newly developed to describe electromagnetism. The term \"geometric algebra\" was repopularized in the 1960s by Hestenes, who advocated its importance to relativistic physics. The scalars and vectors have their usual interpretation, and make up distinct subspaces of a geometric algebra. Bivectors provide a more natural representation of the pseudovector quantities in vector algebra such as oriented area, oriented angle of rotation, torque, angular momentum and the electromagnetic field. A trivector can represent an oriented volume, and so on. An element called a blade may be used to represent a subspace of and orthogonal projections onto that subspace. Rotations and reflections are represented as elements. Unlike a vector algebra, a geometric algebra naturally accommodates any number of dimensions and any quadratic form such as in relativity. Examples of geometric algebras applied in physics include the spacetime algebra (and the less common algebra of physical space) and the conformal geometric algebra. Geometric calculus, an extension of GA that incorporates differentiation and integration, can be used to formulate other theories such as complex analysis and differential geometry, e.g. by using the Clifford algebra instead of differential forms. Geometric algebra has been advocated, most notably by David Hestenes and Chris Doran, as the preferred mathematical framework for physics. Proponents claim that it provides compact and intuitive descriptions in many areas including classical and quantum mechanics, electromagnetic theory and relativity. GA has also found use as a computational tool in computer graphics and robotics."@en . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "En matem\u00E0tiques, \u00E0lgebra geom\u00E8trica \u00E9s un terme aplicat a la teoria de les \u00E0lgebres de Clifford i teories relacionades, seguint un llibre del mateix t\u00EDtol d'Emil Artin. Aquest terme tamb\u00E9 ha tingut recent \u00FAs en els tractaments de la mateixa \u00E0rea en la literatura de f\u00EDsica. Els nombres reals s'utilitzen com escalars en un espai vectorial V. En endavant, un vector \u00E9s una cosa en V mateix. El producte extern (producte exterior, o producte falca) \u2227 es defineix de manera tal que es generi l'\u00E0lgebra graduada (\u00E0lgebra exterior de Hermann Grassmann) de \u039Bn Vn de multivectors. L'\u00E0lgebra geom\u00E8trica \u00E9s l'\u00E0lgebra generada pel producte geom\u00E8tric (el qual \u00E9s pensat com a fonamental) amb (per a tots els multivectors A, B, C): associativitat, distributivitat sobre l'addici\u00F3 de multivectores (A(B + C) = A B + A C y {A + B)C = A C + B C), i la contracci\u00F3 per a qualsevol \u00ABvector\u00BB (un element de grau un) a, a\u00B2 \u00E9s un escalar (nombre real). Anomenem aquesta \u00E0lgebra un \u00E0lgebra geom\u00E8trica . El punt distintiu d'aquesta formulaci\u00F3 \u00E9s la correspond\u00E8ncia natural entre les entitats geom\u00E8triques i els elements de l'\u00E0lgebra associativa. La connexi\u00F3 entre les \u00E0lgebra de Clifford i les formes quadr\u00E0tiques venen de la propietat de contracci\u00F3. Aquesta regla tamb\u00E9 t\u00E9 espai una m\u00E8trica definida pel naturalment derivat producte intern. Ha de ser observat que en \u00E0lgebra geom\u00E8trica en tota la seva generalitat no hi ha restricci\u00F3 cap en el valor de l'escalar; pot succeir que sigui negativa, fins i tot zero (en aquest cas, la possibilitat d'un producte intern est\u00E0 eliminada si es requereix ). El producte escalar usual i el producte creuat tradicional de l'\u00E0lgebra vectorial (a ) troben els seus llocs en l'\u00E0lgebra geom\u00E8trica com el producte intern: (que \u00E9s sim\u00E8tric) I el producte extern: con: (que \u00E9s antisim\u00E8tric). \u00C9s rellevant la distinci\u00F3 entre els vectors axials i polars en l'\u00E0lgebra vectorial, que \u00E9s natural en \u00E0lgebra geom\u00E8trica com la mera distinci\u00F3 entre els vectors i els bivectors (elements de grau dos); i aqu\u00ED \u00E9s la unitat pseudoescalar del 3-espai euclidi\u00E0, el que estableix una dualitat entre els vectors i els bivectors, i se l'anomena aix\u00ED a causa de la propietat prevista i\u00B2 = -1.. Un exemple \u00FAtil \u00E9s , i genera , un cas de l'\u00E0lgebra geom\u00E8trica anomenada \u00E0lgebra de l'espaitemps per Hestenes. El tensor del camp electromagn\u00E8tic, en aquest context, es converteix en simplement un bivector , on la unitat imagin\u00E0ria \u00E9s l'element de volum, donant un exemple de la reinterpretaci\u00F3 geom\u00E8trica dels \"trucs tradicionals\". Els , en aquesta m\u00E8trica de Lorentz, tenen la mateixa expressi\u00F3 que la rotaci\u00F3 en l'espai euclidi\u00E0, on \u00E9s, per descomptat, el bivector generat pel temps i les direccions de l'espai implicades; mentre, que en el cas euclidi\u00E0, \u00E9s el bivector generat per les dues direccions de l'espai, consolidant la \"analogia\" gaireb\u00E9 fins a la identitat."@ca . . . . . . . . . . "Geometric algebra"@en . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "12939"^^ . . . "In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called multivectors. Compared to other formalisms for manipulating geometric objects, geometric algebra is noteworthy for supporting vector division and addition of objects of different dimensions."@en . . . . . . . . . . . "N vector negative.svg"@en . . . . . . . . . . . . . .