Generalizing the Cross Product to N Dimensions: A Novel Approach for Multidimensional Analysis and Applications

Samir Brahim Belhaouari (), Yunis Carreon Kahalan, Ilyasse Aksikas, Abdelouahed Hamdi, Ismael Belhaouari, Elias Nabel Haoudi and Halima Bensmail ()
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Samir Brahim Belhaouari: Computer Sciences and Engineering, Division of Information and Computing Technology, Hamad Bin Khalifa University, Doha P.O. Box 34110, Qatar
Yunis Carreon Kahalan: Computer Sciences and Engineering, Division of Information and Computing Technology, Hamad Bin Khalifa University, Doha P.O. Box 34110, Qatar
Ilyasse Aksikas: Department of Mathematics and Statistics, College of Arts and Sciences, Qatar University, Doha P.O. Box 2713, Qatar
Abdelouahed Hamdi: Department of Mathematics and Statistics, College of Arts and Sciences, Qatar University, Doha P.O. Box 2713, Qatar
Ismael Belhaouari: Department of Advanced Computing Sciences, Maastricht University, P.O. Box 616 Maastricht, The Netherlands
Elias Nabel Haoudi: Program of Science, Texas A&M University-Qatar, Education City, Doha P.O. Box 23874, Qatar
Halima Bensmail: Qatar Computing Research Institute, Qatar Center for Artificial Intelligence, Hamad Bin Khalifa University, Doha P.O. Box 34110, Qatar

Mathematics, 2025, vol. 13, issue 3, 1-23

Abstract: This paper presents a generalization of the cross product to N dimensions, extending the classical operation beyond its traditional confines in three-dimensional space. By redefining the cross product to accommodate N − 1 arguments in N dimensions, a framework has been established that retains the core properties of orthogonality, magnitude, and anticommutativity. The proposed method leverages the determinant approach and introduces the polar sine function to calculate the magnitude of the cross product, linking it directly to the volume of an N -dimensional parallelotope. This generalization not only enriches the theoretical foundation of vector calculus but also opens up new applications in high-dimensional data analysis, machine learning, and multivariate time series. The results suggest that this extension of the cross product could serve as a powerful tool for modeling complex interactions in multi-dimensional spaces, with potential implications across various scientific and engineering disciplines.

Keywords: cross products; linear algebra; determinant; Gram matrix; Gram–Schmidt; orthogonal projections; polar sine; parallelotope (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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