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Repeated vector products

Published online by Cambridge University Press:  08 October 2020

A. F. Beardon  and
N. Lord
A. F. Beardon
Affiliation:
Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd., Cambridge CB3 0WB e-mail: afb@dpmms.cam.ac.uk
N. Lord
Affiliation:
Tonbridge School, Tonbridge, Kent TN9 1JP e-mail: Nick.Lord@tonbridge-school.org
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Extract

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 space3. 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.

Type
Articles
Information
The Mathematical Gazette , Volume 104 , Issue 561 , November 2020 , pp. 460 - 468
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Copyright
© Mathematical Association 2020

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