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| Format: | Artículo científico |
| Language: | en |
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Universidad Nacional Autónoma de México
2004
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| Online Access: | https://www.redalyc.org/articulo.oa?id=40450303 |
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| _version_ | 1866815464915599360 |
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| author | Marco Antonio Murray Lasso |
| author_facet | Marco Antonio Murray Lasso |
| contents | On a Generalization of the Law of Sines to the Tetrahedron and Simplices of Higher Dimensions Marco Antonio Murray Lasso Ingeniería sim plex Law of Sines pro jec tion tet ra he dron vec tor prod uct The well known Law of Sines for tri an gles is care fully an a lyzed to gether with its stan dard proofand based on that anal y sis it is ex tended to the tet ra he dron and simplices of four and more di -men sions. The cru cial step in the proof of the ex ten sion to the tet ra he dron starts by rep r e sent ingeach tri an gle in the skin (sur face) of the tet ra he dron as a vec tor in three-dimensional spac ewhose mag ni tude is equal to the area of the tri an gle and which is nor mal to the plane of the tri -an gle. The sum of these four vec tors is the zero vec tor when the faces are prop erly ori ented. Thenext step is to take the vec tors and pro ject them upon a di rected line that is si mul ta neously per -pen dic u lar to two of the vec tors. The sum of the pro jec tions must be zero, but be cause the di rectedline is or thogo nal to two of them it must also be or thogo nal to the sum of the vec tors rep re sent ingthe two other faces of the tet ra he dron. There are at least two sim ple geo met ri cal in ter pre ta tions tothe main re sult: first, choos ing two dis joint pairs of faces of the tet ra he dron the edge join ing thefirst pair of faces is or thogo nal to the sum of the vec tors rep re sent ing the two other faces; sec ond,the vol umes of two par al lel epi peds formed with trios of vec tor rep re sen ta tion of faces of the tet ra -he dron are equal. The re sult is ex tended to simplices of n di men sions by rep re sent ing the skinof the sim plex by n vec tors or thogo nal to the hyperplanes where the el e ments of the skin lie.Since for n-dimensional spaces it is pos si ble to find a vec tor v simultaneaously or thogo nal ton1 vec tors, the same idea is ap plied and the pro jec tions of the sum of the last two vec tors rep re -sent ing the faces of the skin must be or thogo nal to v. The vec tor prod uct of n 1 vec tors in ndimensional space is used to ob tain v. A sim ple nu mer i cal ex am ple is given. 2004 artículo científico 1405-7743 https://www.redalyc.org/articulo.oa?id=40450303 en http://www.redalyc.org/revista.oa?id=404 Ingeniería. Investigación y Tecnología application/pdf Universidad Nacional Autónoma de México Ingeniería. Investigación y Tecnología (México) Num.3 Vol.V |
| format | Artículo científico |
| id | redalyc_40450303 |
| language | en |
| publishDate | 2004 |
| publisher | Universidad Nacional Autónoma de México |
| spellingShingle | On a Generalization of the Law of Sines to the Tetrahedron and Simplices of Higher Dimensions Marco Antonio Murray Lasso Ingeniería sim plex Law of Sines pro jec tion tet ra he dron vec tor prod uct On a Generalization of the Law of Sines to the Tetrahedron and Simplices of Higher Dimensions Marco Antonio Murray Lasso Ingeniería sim plex Law of Sines pro jec tion tet ra he dron vec tor prod uct The well known Law of Sines for tri an gles is care fully an a lyzed to gether with its stan dard proofand based on that anal y sis it is ex tended to the tet ra he dron and simplices of four and more di -men sions. The cru cial step in the proof of the ex ten sion to the tet ra he dron starts by rep r e sent ingeach tri an gle in the skin (sur face) of the tet ra he dron as a vec tor in three-dimensional spac ewhose mag ni tude is equal to the area of the tri an gle and which is nor mal to the plane of the tri -an gle. The sum of these four vec tors is the zero vec tor when the faces are prop erly ori ented. Thenext step is to take the vec tors and pro ject them upon a di rected line that is si mul ta neously per -pen dic u lar to two of the vec tors. The sum of the pro jec tions must be zero, but be cause the di rectedline is or thogo nal to two of them it must also be or thogo nal to the sum of the vec tors rep re sent ingthe two other faces of the tet ra he dron. There are at least two sim ple geo met ri cal in ter pre ta tions tothe main re sult: first, choos ing two dis joint pairs of faces of the tet ra he dron the edge join ing thefirst pair of faces is or thogo nal to the sum of the vec tors rep re sent ing the two other faces; sec ond,the vol umes of two par al lel epi peds formed with trios of vec tor rep re sen ta tion of faces of the tet ra -he dron are equal. The re sult is ex tended to simplices of n di men sions by rep re sent ing the skinof the sim plex by n vec tors or thogo nal to the hyperplanes where the el e ments of the skin lie.Since for n-dimensional spaces it is pos si ble to find a vec tor v simultaneaously or thogo nal ton1 vec tors, the same idea is ap plied and the pro jec tions of the sum of the last two vec tors rep re -sent ing the faces of the skin must be or thogo nal to v. The vec tor prod uct of n 1 vec tors in ndimensional space is used to ob tain v. A sim ple nu mer i cal ex am ple is given. 2004 artículo científico 1405-7743 https://www.redalyc.org/articulo.oa?id=40450303 en http://www.redalyc.org/revista.oa?id=404 Ingeniería. Investigación y Tecnología application/pdf Universidad Nacional Autónoma de México Ingeniería. Investigación y Tecnología (México) Num.3 Vol.V |
| title | On a Generalization of the Law of Sines to the Tetrahedron and Simplices of Higher Dimensions |
| topic | Ingeniería sim plex Law of Sines pro jec tion tet ra he dron vec tor prod uct |
| url | https://www.redalyc.org/articulo.oa?id=40450303 |