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Cross Product in Euclidean Space with infinite Dimensions

Mathematics Asked by Enock Kabibi on December 19, 2020

How can the cross product of two vectors with infinite tuples be found? ie
If $ A = (a_1, a_2, a_3,…..) $

and $ B = (b_1, b_2, b_3,….) $

One Answer

By definition the cross product of two vectors is a vector that is perpendicular to the two vectors and has length equal to the area of the parallelogram spanned by the vectors (plus orientation). If the dimension is $<3$ such a vector exists iff the given vectors are parallel. In the 3-space it exists always and is unique. If the dimension is $>3$ it exists but is not unique. In an infinite dim space before defining a cross-product you need to define the length of a vector and the area of a parallelogram.

Answered by JCAA on December 19, 2020

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