1,772 research outputs found
A generalization of the parallelogram law to higher dimensions
We propose a generalization of the parallelogram identity in any dimension N 65 2, establishing the ratio of the quadratic mean of the diagonals to the quadratic mean of the faces of a parallelotope. The proof makes use of simple properties of the exterior product of vectors
The vector algebra war: a historical perspective
There are a wide variety of different vector formalisms currently utilized in
engineering and physics. For example, Gibbs' three-vectors, Minkowski
four-vectors, complex spinors in quantum mechanics, quaternions used to
describe rigid body rotations and vectors defined in Clifford geometric
algebra. With such a range of vector formalisms in use, it thus appears that
there is as yet no general agreement on a vector formalism suitable for science
as a whole. This is surprising, in that, one of the primary goals of nineteenth
century science was to suitably describe vectors in three-dimensional space.
This situation has also had the unfortunate consequence of fragmenting
knowledge across many disciplines, and requiring a significant amount of time
and effort in learning the various formalisms. We thus historically review the
development of our various vector systems and conclude that Clifford's
multivectors best fulfills the goal of describing vectorial quantities in three
dimensions and providing a unified vector system for science.Comment: 8 pages, 1 figure, 1 tabl
Further generalizations of the parallelogram law
In a recent work of Alessandro Fonda, a generalization of the parallelogram law in any dimension was given by considering the ratio of the quadratic mean of the measures of the -dimensional diagonals to the quadratic mean of the measures of the faces of a parallelotope. In this paper, we provide a further generalization considering not only -dimensional diagonals and faces, but the -dimensional ones for every
Further generalizations of the parallelogram law
In a recent work of Alessandro Fonda, a generalization of the parallelogram law in any dimension N >= 2 was given by considering the ratio of the quadratic mean of the measures of the (N - 1)-dimensional diagonals to the quadratic mean of the measures of the faces of a parallelotope. In this paper, we provide a further generalization considering not only (N - 1)-dimensional diagonals and faces, but the k-dimensional ones for every 1 <= k <= N - 1
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