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De Vectores Al´Algebra Geométrica De Vectores al Algebra´ Geom´etrica Sergio Ramos Ram´ırez, Jos´eAlfonso Ju´arezGonz´alez, Volkswagen de M´exico 72700 San Lorenzo Almecatla, Cuautlancingo, Pue., M´exico Garret Sobczyk Universidad de las Am´ericas-Puebla Departamento de F´ısico-Matem´aticas 72820 Puebla, Pue., M´exico February 4, 2018 Abstract El Algebra´ Geom´etricaes la extensi´onnatural del concepto de vector y de la suma de vectores. Despu´esde revisar las propiedades de la suma vec- torial, definimos la multiplicaci´onde vectores de tal manera que respete el famoso Teorema de Pit´agoras. Varias demostraciones sint´eticasde teo- remas en geometr´ıaeuclidiana pueden as´ıser reemplazadas por elegantes demostraciones algebraicas. Aunque en este art´ıculonos limitamos a 2 y 3 dimensiones, el Algebra´ Geom´etricaes aplicable a cualquier dimensi´on, tanto en geometr´ıaseuclidianas como no-euclidianas. 0 Introducci´on La evoluci´ondel concepto de n´umero,el cual es una noci´oncentral en Matem´aticas, tiene una larga y fascinate historia, la cual abarca muchos siglos y ha sido tes- tigo del florecimiento y ca´ıdade muchas civilizaciones [4]. En lo que respecta a la introducci´onde los conceptos de n´umerosnegativos e imaginarios, Gauss hac´ıala siguiente observaci´onen 1831: \... este tipo de avances, sin embargo, siempre se han hecho en primera instancia con pasos t´ımidosy cautelosos". En el presente art´ıculo,presentamos al lector las ideas y m´etodos m´aselementales del Algebra´ Geom´etrica,la cual fue descubierta por William Kingdon Clifford (1845-1879) poco antes de su muerte [1], y constituye la generalizaci´onnatural de los sistemas de n´umerosreales y complejos, a trav´esde la introducci´onde nuevos entes matem´aticosdenominados n´umeros con direcci´on. En la Secci´on1, extendemos el sistema de los n´umerosreales R para incluir vectores, los cuales son segmentos de l´ınea con longitud y direcci´on. Dado que el 1 significado geom´etricode la suma vectorial y de la multiplicaci´onde vectores con n´umeros reales o escalares es bien conocido, damos aqu´ıs´oloun breve resumen. Queremos recalcar que el concepto de vector como un segmento dirigido en un espacio plano es independiente de cualquier sistema de coordenadas y de la dimensi´ondel espacio. Es fundamental entender que la posici´on de cada segmento dirigido en un espacio plano no es importante, dado que un vector con origen en un punto dado puede ser trasladado a cualquier otro punto, mientras conserve su longitud y direcci´on. La Secci´on2 trata del producto geom´etrico de vectores. As´ıcomo podemos sumar y multiplicar n´umerosreales, si queremos ampliar el sistema de n´umeros reales para incluir vectores, deberemos ser capaces de multiplicarlos y sumarlos. Como gu´ıa para entender c´omomultiplicar vectores geom´etricamente, recor- damos el milenario Teorema de Pit´agoras,mismo que relaciona los lados de un tri´angulorect´angulo.Dejando de lado ´unicamente la ley universal sobre la conmutatividad de la multiplicaci´on,descubrimos que el producto de dos vec- tores ortogonales es anti-conmutativo y establece una nueva entidad llamada bivector. Definimos los productos interno y exterior en t´erminosde las partes sim´etrica y anti-sim´etrica del producto geom´etricode vectores, e investigamos varias relaciones importantes entre estos tres productos. ´ En la Secci´on3, nos limitamos a las Algebras Geom´etricasm´asb´asicas G2 del 2 3 ´ plano euclidiano R y G3 del espacio euclidiano R . Estas Algebras Geom´etricas ofrecen ejemplos concretos y permiten realizar c´alculosbasados en los conocidos sistemas de coordenadas rectangulares en R2 y R3, aunque debemos tener en cuenta las discusiones generales de las secciones anteriores. A finales del siglo XIX, se libr´ola gran lucha entre el cuaterni´on y el Algebra´ Vectorial de Gibbs- Heaviside [3]. Mostramos c´omoel producto cruz est´andarde dos vectores es el dual natural del producto exterior de esos vectores, as´ıcomo la relaci´oncon otras identidades bien conocidas en el An´alisisVectorial est´andar.Estas ideas pueden generalizarse f´acilmente a Algebras´ Geom´etricasde mayor dimensi´on tanto en espacios euclidianos como no-euclidianos, utilizados ampliamente en las famosas Teor´ıasde la Relatividad de Einstein [5] y en diversos campos de las Matem´aticas[7, 8] y la Ingenier´ıa[2, 6]. En la Secci´on4 abordamos algunas ideas elementales de la Geometr´ıaAnal´ıtica, incluyendo la ecuaci´onvectorial de una l´ınearecta y la de un plano. Presenta- mos f´ormulas para la descomposici´onde un vector en componentes paralela y perpendicular a una l´ıneay a un plano, as´ıcomo expresiones para la reflexi´on y la rotaci´onde un vector en espacios de 2, 3 y m´asdimensiones. En la Secci´on5, la flexibilidad y el poder del Algebra´ Geom´etricase reve- lan completamente al discutir la proyecci´onestereogr´afica de la esfera unitaria en 2D, centrada en el origen, sobre el plano euclidiano. La proyecci´ones- tereogr´afica,y su generalizaci´ona dimensiones mayores, tienen profundas apli- caciones en muchas ´areasde las Matem´aticasy la F´ısica. Por ejemplo, los spinors de 2 componentes, fundamentales en Mec´anicaCu´antica, tienen una interpretaci´ondirecta en la proyecci´onestereogr´aficade la esfera bidimensional [12]. Es notable que a casi 140 a~nosde su descubrimiento, este poderoso sistema 2 num´erico-geom´etrico,el complemento natural del sistema de n´umerosreales que incluye el concepto de direcci´on,no sea universalmente conocido por la comu- nidad cient´ıfica,a pesar de que han habido muchos desarrollos y aplicaciones de esta teor´ıaa niveles avanzados en Matem´aticas, F´ısicaTe´oricay, m´asrecien- temente, en Inform´aticay Rob´otica. Creemos que la raz´onprincipal de esta lamentable situaci´onha sido la falta de una presentaci´onconcisa y rigurosa en los niveles b´asicosde ense~nanza. Por esta raz´on,prestamos mucha atenci´ona la introducci´onde los productos interno y exterior, y desarrollamos las identi- dades b´asicasde una manera clara y directa, de tal forma que su generalizaci´on a Algebras´ Geom´etricaseuclidianas y no-euclidianas de mayor dimensi´onno presente grandes obst´aculos. Finalmente, damos referencias cuidadosamente seleccionadas a material m´asavanzado, que el lector podr´aseguir seg´unsus propios intereses. 1 Suma geom´etricade vectores Los n´umerosnaturales se utilizan para expresar cantidades de objetos, como 3 vacas, 4 libras o 5 pasos hacia el norte. Hist´oricamente, los n´umerosnaturales se han ampliado gradualmente para incluir fracciones, n´umeros negativos y todos los n´umerosde la recta num´erica.Los vectores, o segmentos de l´ınea dirigidos, son un nuevo tipo de n´umeroque incluye la noci´onde direcci´on. Un vector v = jvjv^ tiene longitud jvj y direcci´onunitaria v^, como se muestra en la Figura 1, donde tambi´en se ilustra la suma w = u + v. Sean a, b y c vectores. Cada una de las gr´aficasen la Figura 2 muestra una propiedad geom´etricab´asicade la suma vectorial, junto con su correspondiente traducci´onalgebraica. Por ejemplo, el negativo del vector a es el vector −a, el cual tiene la misma longitud que el vector a pero direcci´on u orientaci´on opuesta, como se muestra en la Figura 2: 1). Procedemos ahora a resumir las leyes algebraicas que rigen la suma geom´etricade vectores, y su multiplicaci´on con n´umerosreales. (A1) a + (−a) = 0a = a0 = 0 Inverso aditivo de un vector (A2) a + b = b + a Ley Conmutativa de la suma vectorial Figure 1: Suma vectorial. 3 Figure 2: Propiedades geom´etricasde la suma vectorial. (A3)( a + b) + c = a + (b + c) := a + b + c Ley Asociativa de la suma vectorial (A4) Para toda α 2 R, αa = aα Los n´umeros reales conmutan con los vectores (A5) a − b := a + (−b) Definici´onde la resta vectorial En la regla (A1), el s´ımbolo 0 representa tanto el vector cero como el escalar cero. La regla (A4) nos indica que la multiplicaci´onde un vector con un n´umero real es una operaci´onconmutativa. Notemos que las reglas para la suma de vec- tores son las mismas que para la suma de n´umerosreales. Aunque los vectores usualmente se representan en t´erminos de alg´unsistema de coordenadas, de- seamos enfatizar que sus propiedades geom´etricasson independientes de ´el.En la Secci´on4 presentamos expl´ıcitamente c´alculosen las Algebras´ Geom´etricas 2 3 G2 y G3, empleando los sistemas de coordenadas ortonormales de R y R , respectivamente. 2 Producto geom´etricode vectores El significado geom´etricode la suma de vectores se representa en las Figuras 1 y 2, y lo formalizamos a trav´esde las reglas (A1) - (A5). Pero, >qu´ehay acerca de la multiplicaci´onentre vectores? Sumamos y multiplicamos n´umerosreales, 4 >por qu´eno podr´ıamoshacer lo mismo con vectores? Veamos c´omomultiplicar vectores de una forma geom´etricamente consistente. Recordemos que, para todo vector, a = jaja^. Elevando al cuadrado esta igualdad, obtenemos a2 = (jaja^)(jaja^) = jaj2a^2 = jaj2: (1) En el ´ultimopaso, hemos introducido una nueva regla que establece que el cuadrado de un vector unitario da como resultado el escalar +1. Esto es siempre cierto para vectores euclidianos unitarios, que son aquellos con los que estamos m´asfamiliarizados.1 Bajo esta premisa se sigue inmediatamente que el cuadrado de un vector euclideano es igual a su magnitud o longitud al cuadrado, a2 = jaj2 ≥ 0, y es cero s´olocuando su longitud es nula. Dividiendo ambos lados de la ecuaci´on(1) por jaj2, obtenemos a2 a a = a = a = 1; (2) jaj2 jaj2 jaj2 o aa−1 = a−1a = 1 donde 1 a a^ a−1 := = = (3) a jaj2 jaj es el inverso multiplicativo del vector a.
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