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If veca,vecb and vecc are three non copl...

If `veca,vecb and vecc` are three non coplanar vectors and `vecr` is any vector in space, then `(vecxxvecb),(vecrxxvecc)+(vecb xxvecc)xx(vecrxxveca)+(veccxxveca)xx(vecrxxvecb)=` (A) `[veca vecb vecc]` (B) `2[veca vecb vecc]vecr` (C) `3[veca vecb vecc]vecr` (D) `4[veca vecb vecc]vecr`

A

`[veca vecb vecc]vecr`

B

`2 [veca vecb vecc]vecr`

C

`3 [veca vecb vecc]vecr`

D

none of these

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Verified by Experts

The correct Answer is:
b
` (veca xx vecb) xx (vecr xx vecc)`
`= ((veca xx vecb) .vecc) vecr- ((veca xx vecb).vecr) .vecc`
` [veca vecb vecc] vecr- [veca vecb vecr] vecc`
similarly, ` (vecb xx vecc) xx (vecr xx veca)`
`= [vecb vecc veca] vecr - [vecb vecc vecr] veca`
` and (vecc xx veca) xx (vecr xx vecb) = [vecc veca vecb] vecr - [vecc veca vecr] vecb`
` Rightarrow (veca xx vecb ) xx (vecr xx vecc) + (vecb xx vecc)`
`x (vecr xx veca) + (vecc xx veca0 xx (vecr xx vecb)`
` 3[ veca vecb vecc] vecr - ([vecb vecc vecr]veca + [vecc veca vecr] vecr + [veca vecb vecr] vecc)`
` 3[veca vecb vecc] vecr - [veca vecb vecc] vecr = 2 [veca vecb vecc] vecr`
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CENGAGE-DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS -Exercise
  1. Let (veca (x) = (sin x) hati+ (cos x) hatj and vecb(x) = (cos 2x) hati...

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  2. For any vectors veca and vecb, (veca xx hati) + (vecb xx hati) + ( vec...

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  3. If veca,vecb and vecc are three non coplanar vectors and vecr is any v...

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  4. If vecP = (vecbxxvecc)/([vecavecbvecc]).vecq=(veccxxveca)/([veca vecb ...

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  5. A (veca), B (vecb) and C (vecc) are the vertices of triangle ABC and R...

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  6. If veca , vecb and vecc are non- coplanar vectors and veca xx vecc is ...

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  7. If V be the volume of a tetrahedron and V ' be the volume of anothe...

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  8. [(veca xxvecb)xx(vecb xx vecc) (vecb xxvecc) xx (vecc xxveca) (veccxxv...

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  9. If vecr=x(1)(vecaxx vecb) + x(2) (vecb xxveca) + x(3)(vecc xxvecd) and...

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  10. If the vectors veca and vecb are perpendicular to each other then a ve...

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  11. If veca,' = hati + hatj, vecb'= hati - hatj + 2hatk nad vecc' = 2hati ...

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  12. If veca= hati +hatj, vecb= hatj + hatk, vecc = hatk + hati then in th...

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  13. If unit vectors veca and vecb are inclined at an angle 2 theta such th...

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  14. vecb and vecc are non- collinear if veca xx (vecb xx vecc) + (veca .ve...

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  15. Unit vectors veca and vecb ar perpendicular , and unit vector vecc is ...

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  16. If vectors veca and vecb are two adjacent sides of parallelograsm then...

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  17. If vecca xx (vec b xx vecc) is perpendicular to (veca xx vecb ) xx vec...

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  18. If vecp=(vecbxxvecc)/([(veca,vecb,vecc)]),vecq=(veccxxveca)/([(veca,ve...

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  19. a(1), a(2),a(3) in R - {0} and + a(1)+ a(2)cos2x+ a(3)sin^(2)x=0 " for...

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  20. If veca and vecb are two vectors and angle between them is theta , the...

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