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,….) $

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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