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A(veca),B(vecb),C(vecc) are the vertices...

`A(veca),B(vecb),C(vecc)` are the vertices of the triangle ABC and `R(vecr)` is any point in the plane of triangle ABC , then `vec r.(vecaxxvecb+vecbxxvecc+veccxxveca)` is always equal to

A

zero

B

` [veca vecb vecc ] `

C

`- [ veca vecb vecc]`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
b
A vector perpendicular to the plane of ` A(veca) , B(vecb) and C(vecc)` is
`(vecb -veca) xx (vecc-veca) = vecaxxvecb +vecb xx vecc +vecc xx veca`
Now for any point `R(vecr)` in the plane of A, B and C is
`(vecr-veca). (veca xx vecb + vecb xx vecc+veccxxveca) =0`
`vecr. (veca xx vecb +vecb xxvecc + vecc xxveca)`
`-veca. (veca xx vecb + vecb xxvecc +veccxxveca)=0`
`vecr. (vecaxxvecb + vecbxxvecc +veccxxveca)`
` vec0 +veca.vecb xx vecc +vec0`
`vecr. (veca xx vecb + vecb xx vecb xx vecc + vecc xx veca0 = [ veca vecb vecc]`
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CENGAGE PUBLICATION-DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS -Exercises MCQ
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  2. If vecP = (vecbxxvecc)/([vecavecbvecc]).vecq=(veccxxveca)/([veca vecb ...

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

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

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  5. If V be the volume of a tetrahedron and V ' be the volume of another...

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  6. [( vec axx vec b)xx( vec bxx vec c)( vec bxx vec c)xx( vec cxx vec a)(...

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

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  8. If veca bot vecb then vector vecv in terms of veca and vecb satisfying...

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  9. If vec a^'= hat i+ hat j , vec b^' hat i- hat j+2 hat ka n d vec c^'2...

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

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  11. If unit vectors vec a and vec b are inclined at angle 2theta such t...

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  12. vec a, vec ba n d vec c are non-collinear if vec axx( vec bxx vec c)+...

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

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  14. If vectors vec aa n d vec b are two adjacent sides of a parallelogr...

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  15. If vec axx( vec bxx vec c) is perpendicular to ( vec axx vec b)xx vec...

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  16. Let vec a , vec b ,a n d vec c be vectors forming right-hand traid. L...

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  17. 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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  18. If vec aa n d vec b are two vectors and angle between them is thet...

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  19. Let veca and vecb be two non- zero perpendicular vectors. A vector vec...

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  20. If vector vec b=(t a nalpha,-1, 2sqrt(sinalpha//2))a n d vec c=(t a n...

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