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Let veca,vecb and vecc be a set of non- ...

Let `veca,vecb and vecc` be a set of non- coplanar vectors and `veca'vecb' and vecc'` be its reciprocal set.
prove that `veca=(vecb'xxvecc')/([veca'vecb'vecc']),vecb=(vecc'xxveca')/([veca'vecb'vecc'])andvecc=(veca'xxvecb')/([veca'vecb'vecc'])`

Text Solution

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we have , `vecb'xxvecc'=((veccxxveca)xx(vecaxxvecb))/([vecavecbvecc]^(2))`
`({(veccxxveca).vecb}veca-{(veccxxveca).veca}vecb)/([vecavecbvecc]^(2))=([vecc vecavecb]veca-[veccvecaveca]vecb)/([veca vecbvecc]^(2))=([vecavecb vecc]veca-0)/([vecavecbvecc]^(2))=(veca)/([veca vecb vecc])`
`[veca'vecb'vecc']=veca'(vecb'xxvecc')=(vecbxxvecc)/([vecavecbvecc]).veca/([vecavecbvecc]^(2))=([veca vecbvecc])/([veca vecbvecc]^(2))=1/([vecavecbvecc])`
`(vecb'xxvecc')/([veca'vecb'vecc'])=veca`
`vecb=(vecc'xxveca')/([veca'vecb'vecc']),vecc=(veca'xxvecb')/([veca'vecb'vecc'])`
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