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Let vec a , vec b ,a n d vec ca n d vec...

Let ` vec a , vec b ,a n d vec ca n d vec a^' , vec b^' , vec c '` are reciprocal system of vectors, then prove that ` vec a^'xx vec b^'+ vec b^'xx vec c^'+ vec c^'xx vec a^'=( vec a+ vec b+ vec c)/([ vec a vec b vec c])` .

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`veca'xxvecb'=((vecbxxvecc)xx(veccxxveca))/([vecavecbvecc]^(2))=({(vecbxxvecc).veca}vecc-{(vecbxxvecc).vecc}veca)/([vecavecbvecc]^(2))=([vecb vecc veca]vecc)/([vecabvecbvecc]^(2))=([veca vecb vecc]vecc)/([veca vecb vecc]^(2))=vecc/([veca vecb vecc])`
similarly, `vecb'xxvecc'=veca/([vecaxxvecbxxvecc])andvecc'xxveca' = vecb/([vecavecbvecc])`
Adding `veca'xxvecb'+vecb'xxvecc'+vecc'xxveca'=(veca+vecb+vecc)/([vecavecbvecc])`
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