Vetores em Geometria Anal\303\255ticaO que segue aqui \303\251 apenas uma amostra de utiliza\303\247\303\243o do Maple V no estudo de vetores em Geometria Anal\303\255tica.
Vetores
Consideramos vetores dados em coordenadas , no plano 2D ou no espa\303\247o 3D , em rela\303\247\303\243o a alguma base ordenada.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Definindo um vetor 2D ou 3D dadas as suas coordenadas: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Definindo um vetor 2D ou 3D dadas as suas coordenadas:
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 um vetor 2D ou 3D literal, para defini\303\247\303\243o posterior das coordenadas: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Depend\303\252ncia e independ\303\252ncia linear
Para analisar depend\303\252ncia e independ\303\252ncia linear, via coordenadas, devemos analisar matrizes constru\303\255das com os vetores:
Se o posto da matriz cujas linhas (ou colunas) s\303\243o as coordenadas dos vetores for exatamente o n\303\272mero de vetores, ent\303\243o os vetores s\303\243o l.i. (lineramente independentes) , caso contr\303\241rio s\303\243o l.d. (linearmente dependentes).
Construindo a matriz cujas linhas [ colunas ] s\303\243o as coordenadas dos vetores: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Obs : stack , concat e augment n\303\243o manipulam somente com vetores. Pode-se misturar com matrizes, desde que de tamanhos coerentes, j\303\241 que tanto vetores como matrizes s\303\243o arrays .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 o posto (caracter\303\255stica) da matriz A
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVElcmFua0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUZHNiVRIkFGJ0ZKRk0vJStleGVjdXRhYmxlR0Y0Ri9GLy1GLDYtUSI7RidGL0YyL0Y2RkxGN0Y5RjtGPUY/RkEvRkVRLDAuMjc3Nzc3OGVtRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGQy8lJmRlcHRoR0Zeby8lKmxpbmVicmVha0dRJWF1dG9GJy1GRzYjUSFGJ0ZXRi8=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYwLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVElcmFua0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUZHNiVRIk1GJ0ZKRk0vJStleGVjdXRhYmxlR0Y0Ri9GLy1GLDYtUSI7RidGL0YyL0Y2RkxGN0Y5RjtGPUY/RkEvRkVRLDAuMjc3Nzc3OGVtRidGKy1GLDYtUSIjRidGL0YyRjVGN0Y5RjtGPUY/RkFGRC1GRzYlUWVvfk9ic2VydmV+cXVlfk1+JiMyMzM7fmF+dHJhbnNwb3N0YX5kZX5BfmV+cG9ydGFudG8sfmRldmV+dGVyfm9+bWVzbW9+cG9zdG9GJ0ZKRk0tSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGQy8lJmRlcHRoR0Zkby8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GYG82JkZib0Zlb0Znby9Gam9RJWF1dG9GJ0Zfb0ZccC1GRzYjUSFGJ0ZXRi8=Lembramos que o posto de A \303\251 o n\303\272mero de linhas n\303\243o nulas de uma matriz escada l-equivalente a A :
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Analisando a depend\303\252ncia linear
Como o n\303\272mero de vetores para compor A era 3 (vetores u, v e w) e o posto de A \303\251 2, os vetores em quest\303\243o s\303\243o l. d. O posto sendo 2, \303\251 poss\303\255vel extrair no m\303\241ximo 2 vetores l. i. do conjunto de 3.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 os dois primeiros s\303\243o l.i ., o terceiro, w, pode ser obtido como c.l. (combina\303\247\303\243o linear) dos dois primeiros.
(*) Ver t\303\263pico abaixo, sobre combina\303\247\303\265es lineares.: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Obs: \303\211 sempre bom lembrar alguns detalhes para evitar de fazer contas desnecess\303\241rias. Por exemplo:
1. Vetor nulo \303\251 combina\303\247\303\243o linear de quaisquer outros vetores. Logo, qualquer conjunto de vetores contendo o vetor nulo entre eles \303\251 sempre l.d .
2. Tr\303\252s ou mais vetores no plano s\303\243o sempre l.d . , assim como quatro ou mais no espa\303\247o 3D. Ou seja, o n\303\272mero m\303\241ximo de vetores l.i . num espa\303\247o \303\251 a dimens\303\243o do espa\303\247o.Combina\303\247\303\243o 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 combina\303\247\303\243o linear dos vetores LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYlLUkjbW5HRiQ2JFEiMUYnL0YzUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnRj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZARj4= , ..., LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYlLUYsNiVRIm5GJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y9RkA= \303\251 um vetor da forma LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEnJiM5NDU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYlLUkjbW5HRiQ2JFEiMUYnRjIvJStleGVjdXRhYmxlR0YxRjJGMi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJy1GLDYlUSJ1RicvRjBRJXRydWVGJy9GM1EnaXRhbGljRidGNUY+RjI=+ ... + 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 , onde \316\261[1] , ..., \316\261[n] s\303\243o escalares .
Vimos acima um exemplo de como obter um vetor como combina\303\247\303\243o linear de outros.
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A resposta acima indica que w = -u + 2v , como podemos checar:
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Produtos interno, vetorial e misto
O produto interno ou escalar de dois vetores \303\251 um escalar . Todas as coordenadas s\303\243o segundo uma base ortonormal de vetores (2 a 2 ortogonais e unit\303\241rios).
O produto vetorial de dois vetores resulta num vetor ortogonal aos dois. As coordenadas devem ser dadas segundo uma base ortonormal positiva do espa\303\247o. \303\211 um produto exclusivo do espa\303\247o 3D.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Observa\303\247\303\243o: o comando "innerprod(u,v)" \303\251 mais utilizado em \303\201lgebra Linear, realizando o produto interno usual de vetores e tamb\303\251m de matrizes.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Veja que o produto vetorial de u com v, w = u x v , \303\251 ortogonal aos vetores u e v. A ortogonalidade de dois vetores \303\251 verificada quando o produto escalar \303\251 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O produto vetorial n\303\243o \303\251 sim\303\251trico :
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 norma (comprimento) de um vetor relativamente a uma base ortonormal \303\251 dado pela raiz quadrada de cada elememto elevado ao quadradoe no Maple V, pode ser expresso na forma sqrt(dotprod(u,u)) ou diretamente, usando norm(u,2 ) .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 u . v = ||u|| ||v|| cos \316\270 , onde \316\270 \303\251 o \303\242ngulo entre u e v, temos que
\316\270 = arccos( u.v / (||u|| ||v||) ),
exceto quando um dos vetores \303\251 nulo. O Maple V possui a fun\303\247\303\243o angle .
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Produto misto: (u x v). w = [u, v, w] . Vamos denotar, para nosso uso, misto(u, v, w), que pode ser calculado atrav\303\251s do determinante da matriz cujas linhas s\303\243o u, v, e w. Vamos definir uma fun\303\247\303\243o para calcular o produto misto: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 troca de ordem dos vetores no produto misto pode alterar ou n\303\243o o sinal, conforme se altera ou n\303\243o a orienta\303\247\303\243o (positiva ou negativa) do conjunto. Quando {u, v, w} forma uma base positiva do espa\303\247o, [u, v, w] > 0 e representa o volume do paralelep\303\255pedo de lados u, v e w.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