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Prove that [veclvecmvecn][vecavecbvecc]=...

Prove that `[veclvecmvecn][vecavecbvecc]=|{:(vecl.veca,vecl.vecb,vecl.vecc),(vecm.veca,vecm.vecb,vecm.vecc),(vecn.veca,vecn.vecb,vecn.vecc):}|`

Text Solution

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Let `vecl=l_(1)hati+l_(2)hatj+l_(3)hatk,vecm=m_(1)hati+m_(2)hatj+m_(3)hatk andvecn=n_(1)hati+n_(2)hatj+n_(3)hatk`
`veca=a_(1)hati+a_(2)hatj=a_(3)hatk,vecb=b_(1)hati+b_(2)hatj+b_(3)hatk nandvecc=c_(1)hati+c_(2)hatj+c_(3)hatk`
`vecl.veca=l_(1)a_(1)+l_(2)a_(2)+l_(3)a_(3)=suml_(1)a_(1)`
similarly `vecl.vecb=suml_(1)b_(1).etc.`
`[veclvecmvecn][vecavecbvecc]=|{:(l_(1),l_(2),l_(3)),(m_(1),m_(2),m_(3)),(n_(1),n_(2),n_(3)):}||{:(a_(1),a_(2),a_(3)),(b_(1),b_(2),b_(3)),(c_(1),c_(2),c_(3)):}|`
`=|{:(suml_(1)a_(1),suml_(1)b_(1),suml_(1)c_(1)),(summ_(1)a_(1),summ_(2)b_(1),summ_(1)c_(1)),(sumn_(1)a_(1),sumn_(1)b_(1),sumn_(1)c_(1)):}|`
`=|{:(vecl.veca,vecl.vecb,veca.vecc),(vecm.veca,vecm.vecb,vecm.vecc),(vecn.veca,vecn.vecb,vecn.vecc):}|`
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