SIFAT-SIFAT HASIL KALI SILANG DI RUANG EUCLID DIMENSI N

SUGIANTO, . (2025) SIFAT-SIFAT HASIL KALI SILANG DI RUANG EUCLID DIMENSI N. Sarjana thesis, UNIVERSITAS NEGERI JAKARTA.

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Abstract

Penelitian ini mengkaji sifat-sifat operasi hasil kali silang, yang umumnya dikenal di ruang tiga dimensi, pada ruang matematika dengan dimensi yang lebih tinggi (ruang Euclid dimensi $n$). Latar belakang studi menunjukkan bahwa meskipun konsep ruang vektor sangat penting dalam matematika dan berbagai bidang terapan, generalisasi hasil kali silang ke dimensi yang lebih tinggi masih memerlukan eksplorasi. Adanya aplikasi terkait seperti metode Gram-Schmidt dan aturan Cramer menunjukkan relevansi topik ini. Oleh karena itu, penelitian ini bertujuan untuk mengidentifikasi bagaimana sifat-sifat hasil kali silang berlaku di $\mathbb{R}^n$ dan bagaimana operasi ini bekerja ketika diterapkan pada vektor satuan standar. Hasil analisis menunjukkan bahwa banyak sifat dasar hasil kali silang dari ruang tiga dimensi tetap berlaku di $\mathbb{R}^n$, termasuk sifat distributif dan perkalian dengan skalar. Namun, ditemukan bahwa sifat antikomutatif (urutan operasi membalik tanda hasil) tidak selalu berlaku di $\mathbb{R}^n$ pada kondisi tertentu, karena pembalikan urutan vektor tidak selalu mengubah tanda hasil. Selanjutnya, hasil kali silang $n$-$1$ vektor satuan standar secara berurutan menghasilkan pola yang tergantung pada nilai $n$ (modulo 4), di mana tanda vektor hasil bervariasi antara positif, mengikuti tanda kofaktor, atau bahkan tetap sama saat urutan operasi dibalik. ***** This study investigates the properties of the cross product operation, commonly known in three-dimensional space, in a higher-dimensional mathe�matical space ($n$-dimensional Euclidean space). The background of the study indicates that although the concept of vector spaces is crucial in mathematics and various applied fields, the generalization of the cross product to higher dimensions still requires exploration. The existence of related applications such as the Gram-Schmidt method and Cramer’s rule demonstrates the relevance of this topic. Therefore, this study aims to identify how the properties of the cross product hold in $n$-dimensional space and how this operation works when applied to standard unit vectors. The analysis shows that many basic properties of the cross product of three-dimensional space hold in $\mathbb{R}^n$, including the distributive property and multiplication by a scalar. However, it is found that the anticommutative property (the order of operations reverses the sign of the result) does not always hold in Rn under certain conditions, because reversing the order of vectors does not always change the sign of the result. Furthermore, successive cross products of $n$-$1$ standard unit vectors produce a pattern that depends on the value of $n$ (modulo 4), where the sign of the result vector varies between being positive, following the sign of the cofactor, or even remaining the same when the order of operations is reversed.

Item Type:	Thesis (Sarjana)
Additional Information:	1). Dr. Yudi Mahatma, M.Si. 2). Ibnu Hadi, M.Si.
Subjects:	Sains > Matematika
Divisions:	FMIPA > S1 Matematika
Depositing User:	Sugianto .
Date Deposited:	19 Aug 2025 03:14
Last Modified:	19 Aug 2025 03:14
URI:	http://repository.unj.ac.id/id/eprint/61543

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