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Algebraic and geometric definitions of the cross product: the link

Published online by Cambridge University Press:  12 November 2024

François Dubeau
François Dubeau*
Affiliation:
Département de mathématiques, Faculté des sciences, Université de Sherbrooke 2500, boul. de l’Université, Sherbrooke (Qc), Canada, J1K 2R1 e-mail: francois.dubeau@usherbrooke.ca
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Given two vectors $\overrightarrow u= {({u_1},\,\,{u_2},\,{u_3})^t}$ and $\overrightarrow y= {({v_1},\,{v_2},\,{v_3})^t}$ in ${{\mathcal{R}}^3}$, the cross product $\overrightarrow u\times \overrightarrow v $is defined as follows (see [1] or [2]):

$$Algebraic{\rm{ }}definition{\rm{:}}\,\,\overrightarrow u\times \overrightarrow v \, = \,\left[ {\matrix{ {{u_2}{v_3}} \hfill &-\hfill & {{u_3}{v_2}} \hfill\cr{{u_3}{v_1}} \hfill &-\hfill & {{u_1}{v_3}} \hfill\cr{{u_1}{v_2}} \hfill &-\hfill & {{u_2}{v_1}} \hfill\cr} } \right].$$

Type
Articles
Information
The Mathematical Gazette , Volume 108 , Issue 573 , November 2024 , pp. 470 - 477
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Copyright
© The Authors, 2024 Published by Cambridge University Press on behalf of The Mathematical Association

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