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Posiciones relativas de dos planos - Historial de revisiones
2024-03-29T06:38:37Z
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Laura.2mdc en 08:10 6 ago 2008
2008-08-06T08:10:36Z
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<td colspan='2' style="background-color: white; color:black;">Revisión de 08:10 6 ago 2008</td>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">{{revisar}}</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Introducción==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Introducción==</div></td></tr>
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Laura.2mdc
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Laura.2mdc: ortografía
2008-08-06T08:09:14Z
<p>ortografía</p>
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<td colspan='2' style="background-color: white; color:black;">← Revisión anterior</td>
<td colspan='2' style="background-color: white; color:black;">Revisión de 08:09 6 ago 2008</td>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>==<del style="color: red; font-weight: bold; text-decoration: none;">Introduccion</del>==</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>==<ins style="color: red; font-weight: bold; text-decoration: none;">Introducción</ins>==</div></td></tr>
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Laura.2mdc
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83.61.41.73 en 13:02 19 dic 2006
2006-12-19T13:02:45Z
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<td colspan='2' style="background-color: white; color:black;">Revisión de 13:02 19 dic 2006</td>
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<tr><td colspan="2" class="diff-lineno">Línea 61:</td>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> a_2 & b_2 & c_2</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> a_2 & b_2 & c_2</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> \end{array}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> \end{array}</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\right)\<del style="color: red; font-weight: bold; text-decoration: none;">,</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>\right)</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"></math></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>\<ins style="color: red; font-weight: bold; text-decoration: none;">qquad \mathrm{y} \qquad </ins>A | B \, = \,</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"></center></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><center></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><math></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>A | B \, = \,</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\left(</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\left(</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> \left.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> \left.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Según el teorema de Rouché-<del style="color: red; font-weight: bold; text-decoration: none;">Frobenius</del>, se pueden presentar los casos que pasamos a</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Según el teorema de Rouché-<ins style="color: red; font-weight: bold; text-decoration: none;">Fröbenius</ins>, se pueden presentar los casos que pasamos a</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>discutir en la siguiente seccion:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>discutir en la siguiente seccion:</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>El sistema de ecuaciones es compatible indeterminado y tiene infinitas soluciones. Los</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>El sistema de ecuaciones es compatible indeterminado y tiene infinitas soluciones. Los</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>planos se cortan según una recta. Son '''''planos secantes'''''<del style="color: red; font-weight: bold; text-decoration: none;">.</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>planos se cortan según una recta. Son '''''planos secantes<ins style="color: red; font-weight: bold; text-decoration: none;">.</ins>'''''</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td colspan="2" class="diff-lineno">Línea 142:</td>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>El sistema de ecuaciones es compatible indeterminado, siendo la segunda ecuación</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>El sistema de ecuaciones es compatible indeterminado, siendo la segunda ecuación</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>proporcional a la primera. Los planos tienen en comun todos sus puntos. Son '''''planos coincidentes'''''<del style="color: red; font-weight: bold; text-decoration: none;">.</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>proporcional a la primera. Los planos tienen en comun todos sus puntos. Son '''''planos coincidentes<ins style="color: red; font-weight: bold; text-decoration: none;">.</ins>'''''</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>El sistema de ecuaciones es incompatible, no tiene solucion. Los</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>El sistema de ecuaciones es incompatible, no tiene solucion. Los</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>planos no tienen ningun punto en común. Son '''''planos paralelos'''''<del style="color: red; font-weight: bold; text-decoration: none;">.</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>planos no tienen ningun punto en común. Son '''''planos paralelos<ins style="color: red; font-weight: bold; text-decoration: none;">.</ins>'''''</div></td></tr>
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83.61.41.73
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213.97.196.225: /* Introduccion */
2006-12-19T08:27:59Z
<p><span class="autocomment">Introduccion</span></p>
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<td colspan='2' style="background-color: white; color:black;">Revisión de 08:27 19 dic 2006</td>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> a_2 & b_2 & c_2</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> a_2 & b_2 & c_2</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> \end{array}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> \end{array}</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\right)</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>\right)<ins style="color: red; font-weight: bold; text-decoration: none;">\,</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td></tr>
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213.97.196.225
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Fjmolina en 00:46 19 dic 2006
2006-12-19T00:46:21Z
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<td colspan='2' style="background-color: white; color:black;">Revisión de 00:46 19 dic 2006</td>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>===Secantes: &nbsp &nbsp Rango ( A ) = Rango ( A | B ) = 2===</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>===Secantes: &nbsp<ins style="color: red; font-weight: bold; text-decoration: none;">; </ins>&nbsp<ins style="color: red; font-weight: bold; text-decoration: none;">; </ins>Rango ( A ) = Rango ( A | B ) = 2===</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>===Coincidentes: &nbsp &nbsp Rango ( A ) = Rango ( A | B ) = 1===</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>===Coincidentes: &nbsp<ins style="color: red; font-weight: bold; text-decoration: none;">; </ins>&nbsp<ins style="color: red; font-weight: bold; text-decoration: none;">; </ins>Rango ( A ) = Rango ( A | B ) = 1===</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>===Paralelos: &nbsp &nbsp Rango ( A ) = 1, &nbsp; &nbsp; Rango ( A | B ) = 2===</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>===Paralelos: &nbsp<ins style="color: red; font-weight: bold; text-decoration: none;">; </ins>&nbsp<ins style="color: red; font-weight: bold; text-decoration: none;">; </ins>Rango ( A ) = 1, &nbsp; &nbsp; Rango ( A | B ) = 2===</div></td></tr>
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Fjmolina
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Fjmolina en 00:38 19 dic 2006
2006-12-19T00:38:41Z
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<td colspan='2' style="background-color: white; color:black;">← Revisión anterior</td>
<td colspan='2' style="background-color: white; color:black;">Revisión de 00:38 19 dic 2006</td>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>===Rango ( A ) = Rango ( A | B ) = 2===</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>===<ins style="color: red; font-weight: bold; text-decoration: none;">Secantes: &nbsp &nbsp </ins>Rango ( A ) = Rango ( A | B ) = 2===</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td colspan="2" class="diff-lineno">Línea 137:</td>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>===Rango ( A ) = Rango ( A | B ) = 1===</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>===<ins style="color: red; font-weight: bold; text-decoration: none;">Coincidentes: &nbsp &nbsp </ins>Rango ( A ) = Rango ( A | B ) = 1===</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
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<td colspan="2" class="diff-lineno">Línea 176:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>===Rango ( A ) = 1, &nbsp; &nbsp; Rango ( A | B ) = 2===</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>===<ins style="color: red; font-weight: bold; text-decoration: none;">Paralelos: &nbsp &nbsp </ins>Rango ( A ) = 1, &nbsp; &nbsp; Rango ( A | B ) = 2===</div></td></tr>
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Fjmolina
http://www.wikillerato.org/index.php?title=Posiciones_relativas_de_dos_planos&diff=2391&oldid=prev
Fjmolina en 00:36 19 dic 2006
2006-12-19T00:36:34Z
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<td colspan='2' style="background-color: white; color:black;">← Revisión anterior</td>
<td colspan='2' style="background-color: white; color:black;">Revisión de 00:36 19 dic 2006</td>
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<tr><td colspan="2" class="diff-lineno">Línea 1:</td>
<td colspan="2" class="diff-lineno">Línea 1:</td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">==Introduccion==</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><br/></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Dos planos pueden adoptar tres posiciones relativas en el espacio:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Dos planos pueden adoptar tres posiciones relativas en el espacio:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><span</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>1. Secantes.</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">style = 'color:#00aa00'></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>1.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"></span></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Secantes.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><span</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>2. Coincidentes.</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">style = 'color:#00aa00'></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>2.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"></span></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Coincidentes.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><span</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>3. Paralelos.</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">style = 'color:#00aa00'></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>3.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"></span></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Paralelos.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td colspan="2" class="diff-lineno">Línea 97:</td>
<td colspan="2" class="diff-lineno">Línea 89:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Según el teorema de Rouché-Frobenius, se pueden presentar los <del style="color: red; font-weight: bold; text-decoration: none;">siguientes </del>casos:</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Según el teorema de Rouché-Frobenius, se pueden presentar los casos <ins style="color: red; font-weight: bold; text-decoration: none;">que pasamos a</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">discutir en la siguiente seccion</ins>:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><span</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>=<ins style="color: red; font-weight: bold; text-decoration: none;">=Casos que se pueden dar</ins>:<ins style="color: red; font-weight: bold; text-decoration: none;">==</ins></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">style </del>= <del style="color: red; font-weight: bold; text-decoration: none;">'color</del>:<del style="color: red; font-weight: bold; text-decoration: none;">#00aa00'></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">Caso 1:</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><<ins style="color: red; font-weight: bold; text-decoration: none;">br</ins>/></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div></<del style="color: red; font-weight: bold; text-decoration: none;">span</del>></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">&nbsp; &nbsp; &nbsp;</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>=<ins style="color: red; font-weight: bold; text-decoration: none;">==</ins>Rango ( A ) = Rango ( A | B ) = 2<ins style="color: red; font-weight: bold; text-decoration: none;">===</ins></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><span</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">style </del>= <del style="color: red; font-weight: bold; text-decoration: none;">'color:#00aaaa'></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Rango ( A ) = Rango ( A | B ) = 2</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"></span></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td colspan="2" class="diff-lineno">Línea 148:</td>
<td colspan="2" class="diff-lineno">Línea 137:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">----</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>==<ins style="color: red; font-weight: bold; text-decoration: none;">=</ins>Rango ( A ) = Rango ( A | B ) = 1<ins style="color: red; font-weight: bold; text-decoration: none;">===</ins></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><span</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">style </del>= <del style="color: red; font-weight: bold; text-decoration: none;">'color:#00aa00'></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">Caso 2:</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"></span></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">&nbsp; &nbsp; &nbsp;</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><span</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">style </del>= <del style="color: red; font-weight: bold; text-decoration: none;">'color:#00aaaa'></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Rango ( A ) = Rango ( A | B ) = 1</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"></span></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
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<tr><td colspan="2" class="diff-lineno">Línea 199:</td>
<td colspan="2" class="diff-lineno">Línea 176:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">----</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>=<ins style="color: red; font-weight: bold; text-decoration: none;">==Rango ( A ) = 1, </ins>&nbsp; &nbsp; Rango ( A | B ) = 2<ins style="color: red; font-weight: bold; text-decoration: none;">===</ins></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br/></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><span</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">style </del>= <del style="color: red; font-weight: bold; text-decoration: none;">'color:#00aa00'></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">Caso 3:</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"></span></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>&nbsp; &nbsp; <del style="color: red; font-weight: bold; text-decoration: none;">&nbsp;</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><span</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">style = 'color:#00aaaa'></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">Rango ( A ) = </del>Rango ( A | B ) = 2</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"></span></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
</table>
Fjmolina
http://www.wikillerato.org/index.php?title=Posiciones_relativas_de_dos_planos&diff=2385&oldid=prev
Fjmolina en 18:18 18 dic 2006
2006-12-18T18:18:24Z
<p></p>
<table style="background-color: white; color:black;">
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr>
<td colspan='2' style="background-color: white; color:black;">← Revisión anterior</td>
<td colspan='2' style="background-color: white; color:black;">Revisión de 18:18 18 dic 2006</td>
</tr>
<tr><td colspan="2" class="diff-lineno">Línea 29:</td>
<td colspan="2" class="diff-lineno">Línea 29:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Sean dos planos &nbsp;</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Sean dos planos &nbsp;</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\<del style="color: red; font-weight: bold; text-decoration: none;">p21</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>\<ins style="color: red; font-weight: bold; text-decoration: none;">pi_1</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>&nbsp; y &nbsp;</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>&nbsp; y &nbsp;</div></td></tr>
</table>
Fjmolina
http://www.wikillerato.org/index.php?title=Posiciones_relativas_de_dos_planos&diff=2384&oldid=prev
Fjmolina en 18:17 18 dic 2006
2006-12-18T18:17:32Z
<p></p>
<table style="background-color: white; color:black;">
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr>
<td colspan='2' style="background-color: white; color:black;">← Revisión anterior</td>
<td colspan='2' style="background-color: white; color:black;">Revisión de 18:17 18 dic 2006</td>
</tr>
<tr><td colspan="2" class="diff-lineno">Línea 29:</td>
<td colspan="2" class="diff-lineno">Línea 29:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Sean dos planos &nbsp;</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Sean dos planos &nbsp;</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\<del style="color: red; font-weight: bold; text-decoration: none;">pi_1</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>\<ins style="color: red; font-weight: bold; text-decoration: none;">p21</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>&nbsp; y &nbsp;</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>&nbsp; y &nbsp;</div></td></tr>
<tr><td colspan="2" class="diff-lineno">Línea 145:</td>
<td colspan="2" class="diff-lineno">Línea 145:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Rango ( A ) = Rango ( A | B ) = 2</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Rango ( A ) = Rango ( A | B ) = 2</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></center></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></center></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><br/></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">----</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td colspan="2" class="diff-lineno">Línea 192:</td>
<td colspan="2" class="diff-lineno">Línea 196:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Rango ( A ) = Rango ( A | B ) = 1</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Rango ( A ) = Rango ( A | B ) = 1</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></center></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></center></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><br/></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">----</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
</table>
Fjmolina
http://www.wikillerato.org/index.php?title=Posiciones_relativas_de_dos_planos&diff=2383&oldid=prev
Fjmolina en 18:07 18 dic 2006
2006-12-18T18:07:20Z
<p></p>
<p><b>Página nueva</b></p><div>Dos planos pueden adoptar tres posiciones relativas en el espacio:<br />
<br />
<br/><br />
<br />
<span<br />
style = 'color:#00aa00'><br />
1.<br />
</span><br />
Secantes.<br />
<br />
<br/><br />
<br />
<span<br />
style = 'color:#00aa00'><br />
2.<br />
</span><br />
Coincidentes.<br />
<br />
<br/><br />
<br />
<span<br />
style = 'color:#00aa00'><br />
3.<br />
</span><br />
Paralelos.<br />
<br />
<br/><br />
<br />
Sean dos planos &nbsp;<br />
<math><br />
\pi_1<br />
</math><br />
&nbsp; y &nbsp;<br />
<math><br />
\pi_2<br />
</math><br />
&nbsp; de ecuaciones:<br />
<br />
<br/><br />
<br />
<center><br />
<math><br />
\pi_1: \, a_1 x \, + \, b_1 y \, + \, c_1 \, + \, d_1 \, = \, 0<br />
</math><br />
</center><br />
<br />
<br/><br />
<br />
<center><br />
<math><br />
\pi_2: \, a_2 x \, + \, b_2 y \, + \, c_2 \, + \, d_2 \, = \, 0<br />
</math><br />
</center><br />
<br />
<br/><br />
<br />
Para determinar sus posiciones relativas, analizamos el sistema formado por las<br />
ecuaciones de los dos planos, cuyas matrices asociadas son:<br />
<br />
<br/><br />
<br />
<center><br />
<math><br />
A \, = \,<br />
\left(<br />
\begin{array}[c]{ccc}<br />
a_1 & b_1 & c_1<br />
\\<br />
a_2 & b_2 & c_2<br />
\end{array}<br />
\right)<br />
</math><br />
</center><br />
<br />
<br/><br />
<br />
<center><br />
<math><br />
A | B \, = \,<br />
\left(<br />
\left.<br />
\begin{array}[c]{ccc}<br />
a_1 & b_1 & c_1<br />
\\<br />
a_2 & b_2 & c_2<br />
\end{array}<br />
\right|<br />
\begin{array}[c]{c}<br />
-d_1<br />
\\<br />
-d_2<br />
\end{array}<br />
\right)<br />
</math><br />
</center><br />
<br />
<br/><br />
<br />
Según el teorema de Rouché-Frobenius, se pueden presentar los siguientes casos:<br />
<br />
<br/><br />
<br />
<span<br />
style = 'color:#00aa00'><br />
Caso 1:<br />
</span><br />
&nbsp; &nbsp; &nbsp;<br />
<span<br />
style = 'color:#00aaaa'><br />
Rango ( A ) = Rango ( A | B ) = 2<br />
</span><br />
<br />
<br/><br />
<br />
El sistema de ecuaciones es compatible indeterminado y tiene infinitas soluciones. Los<br />
planos se cortan según una recta. Son '''''planos secantes'''''.<br />
<br />
<br/><br />
<br />
Asi, los planos<br />
<br />
<br/><br />
<br />
<center><br />
<math><br />
\pi_1: \, x \, + \, y \, - \, z \, = \, 1<br />
</math><br />
</center><br />
<br />
<br/><br />
<br />
<center><br />
<math><br />
\pi_2: \, x \, + \, y \, + \, z \, = \, 2<br />
</math><br />
</center><br />
<br />
<br/><br />
<br />
son secantes, pues:<br />
<br />
<br/><br />
<br />
<center><br />
Rango ( A ) = Rango ( A | B ) = 2<br />
</center><br />
<br />
<br/><br />
<br />
<span<br />
style = 'color:#00aa00'><br />
Caso 2:<br />
</span><br />
&nbsp; &nbsp; &nbsp;<br />
<span<br />
style = 'color:#00aaaa'><br />
Rango ( A ) = Rango ( A | B ) = 1<br />
</span><br />
<br />
<br/><br />
<br />
El sistema de ecuaciones es compatible indeterminado, siendo la segunda ecuación<br />
proporcional a la primera. Los planos tienen en comun todos sus puntos. Son '''''planos coincidentes'''''.<br />
<br />
<br/><br />
<br />
Asi, los planos<br />
<br />
<br/><br />
<br />
<center><br />
<math><br />
\pi_1: \, x \, + \, y \, - \, z \, = \, 1<br />
</math><br />
</center><br />
<br />
<br/><br />
<br />
<center><br />
<math><br />
\pi_2: \, 2x \, + \, 2y \, - \, 2z \, = \, 2<br />
</math><br />
</center><br />
<br />
<br/><br />
<br />
son coincidentes, pues:<br />
<br />
<br/><br />
<br />
<center><br />
Rango ( A ) = Rango ( A | B ) = 1<br />
</center><br />
<br />
<br/><br />
<br />
<span<br />
style = 'color:#00aa00'><br />
Caso 3:<br />
</span><br />
&nbsp; &nbsp; &nbsp;<br />
<span<br />
style = 'color:#00aaaa'><br />
Rango ( A ) = Rango ( A | B ) = 2<br />
</span><br />
<br />
<br/><br />
<br />
El sistema de ecuaciones es incompatible, no tiene solucion. Los<br />
planos no tienen ningun punto en común. Son '''''planos paralelos'''''.<br />
<br />
<br/><br />
<br />
Asi, los planos<br />
<br />
<br/><br />
<br />
<center><br />
<math><br />
\pi_1: \, x \, + \, y \, - \, z \, = \, 1<br />
</math><br />
</center><br />
<br />
<br/><br />
<br />
<center><br />
<math><br />
\pi_2: \, x \, + \, y \, - \, z \, = \, 2<br />
</math><br />
</center><br />
<br />
<br/><br />
<br />
son paralelos, pues:<br />
<br />
<br/><br />
<br />
<center><br />
Rango ( A ) = 1 &nbsp; &nbsp; &nbsp; mientras que &nbsp; &nbsp; &nbsp; Rango ( A | B ) = 2<br />
</center><br />
<br />
<br/><br />
<br />
[[Category:Matemáticas]]</div>
Fjmolina