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If vectors veca, vecb and vecc are copla...

If vectors `veca, vecb and vecc` are coplanar, show that `|{:(veca, vecb,vecc),(veca.veca,veca.vecb,veca.vecc),(vecb.veca,vecb.vecb,vecb.vecc):}|=vec0`

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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`
` vecn=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,` therefore,
`vecl.veca =l_(1)a_(1)+ l_(2)a_(2) +l_(3)a_(3) = sum l_(1) a_(1) `
similarly `vecl.vecb = suml_(1)b_(1)`.etc.
`now, [vecl vecm vecn] (vecaxxvecb) = |{:(l_(1),l_(2),l_(3)),(m_(1),m_(2),m_(3)),(n_(1),n_(2),n_(3)):}|xx|{:(hati,hatj ,hatk),(a_(1),a_(2),a_(3)),(b_(1),b_(2),b_(3)):}|`
`|{:(suml_(1)hati,suml_(1)a_(1),suml_(1)b_(1)),(summ_(1)hati,summ_(1)a_(1),summ_(1)b_(1)),(sumn_(1)hati, sumn_(1)a_(1),sumn_(1)b_(1)):}|`
`=|{:(vecl,vecl.veca,vecl.vecb),(vecm,vecm.veca,vecm.vecb),(vecn,vecn.veca,vecn.vecb):}|=|{:(vecl.veca,vecl.vecb,vecl),(vecm.veca,vecm.vecb,vecm),(vecn.veca,vecn.vecb,vecn):}|`
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CENGAGE-DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS -Exercise
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  2. If veca , vecb , vecc and vecd are four non-coplanar unit vectors such...

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  3. If vectors veca, vecb and vecc are coplanar, show that |{:(veca, vecb,...

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  4. If the volume of a parallelepiped whose adjacent edges are vec a=2...

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  5. If veca=hati+hatj+hatk and vecb = hati-2hatj+hatk then find vector vec...

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  6. If vecx.veca=0 , vecx.vecb=0 , vecx.vecc=0 and vecxnevec0 ...

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  7. If veca=hati+hatj+hatk and vecb = hati-2hatj+hatk then find vector vec...

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  8. If veca , vecb and vecc are three vectors such that vecaxx vecb =vecc...

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  9. If veca= vecP + vecq, vecP xx vecb = vec0 and vecq. vecb =0 then prove...

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  10. prove that (veca.(vecbxxhati))hati+(veca.(vecbxxhatj))hatj+ (veca.(vec...

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  11. for any four vectors veca,vecb, vecc and vecd prove that vecd. (vecax...

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  12. If veca and vecb be two non-collinear unit vectors such that vecaxx(ve...

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  13. If vecaxx(vecbxxvecc)=(vecaxxvecb)xxvecc for non coplanar veca,vecb,v...

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  14. Let veca,vecb and vecc be the non zero vectors such that (vecaxxvecb)x...

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  15. If vec p , vec q , vec r denote vector vec bxx vec c , vec cxx vec a...

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  16. Let veca, vecb , vecc be non -coplanar vectors and let equations veca'...

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  17. Given unit vectors hatm hatn and hatp such that angle between hatm and...

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  18. Let vec a , vec b ,a n d vec c be non-coplanar unit vectors, equal...

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  19. Let veca=a(1)hati+a(2)hatj+a(3)hatk,vecb=b(1)hati + b(2)hatj + b(3)hat...

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  20. If |(a-x)^2(a-y)^2(a-z)^2(b-x)^2(b-y)^2(b-z)^2(c-x)^2(c-y)^2(c-a)^2|=0...

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