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

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.