Repeated vector products. Issue 561 (November 2020)

• Record Type: Journal Article
• Authors: Beardon, A. F.; Lord, N.
• Is Part Of: Mathematical gazette. Volume 104:Issue 561(2020)
• Page Start: 460
• Page End: 468
• Publication Date: 2020-11
• Subjects: Mathematics -- Periodicals; Mathématique; Mathematik; Mathematics; Periodicals; Ressource Internet (Descripteur de forme); Périodique électronique (Descripteur de forme); Zeitschrift; Online-Publikation; 510
• DOI: 10.1017/mag.2020.100
• Languages: English
• ISSNs: 0025-5572

Abstract

Abstract : In [1 ] the second author observed that it is possible to have a binary operation * on a set X with the property that two different arrangements of brackets in a given combination x1 * … * xn of elements of X yield the same outcome for all choices of the xj . For example, for the operation of subtraction on the set of real numbers, we have $$\left[ {a - \left( {b - c} \right)} \right] - d = a - \left[ {b - \left( {c - d} \right)} \right]$$ for all real numbers a, b, c and d . The author then asked whether or not a similar example might hold for an n -fold vector product on three-dimensional Euclidean space 3 . We shall show here that no such example can exist; thus two different arrangements of brackets in a repeated vector product will, for some vectors, yield different answers.