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Published online by Cambridge University Press: 12 November 2024
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]):
$$Algebraic{\rm{ }}definition{\rm{:}}\,\,\overrightarrow u\times \overrightarrow v \, = \,\left[ {\matrix{ {{u_2}{v_3}} \hfill &-\hfill & {{u_3}{v_2}} \hfill\cr{{u_3}{v_1}} \hfill &-\hfill & {{u_1}{v_3}} \hfill\cr{{u_1}{v_2}} \hfill &-\hfill & {{u_2}{v_1}} \hfill\cr} } \right].$$