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A( vec a),B( vec b)a n dC( vec c) are th...

`A( vec a),B( vec b)a n dC( vec c)` are the vertices of triangle `A B C` and `R( vec r)` is any point in the plane of triangle `A B C ,` then`rdot( vec axx vec b+ vec bxx vec c+ vec cxx vec a)` is always equal to a. zero b. `[ vec a vec b vec c]` c. `-[ vec a vec b vec c]` d. none of these

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 ENGLISH-DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS -Exercises MCQ
  1. If veca,vecb and vecc are three non coplanar vectors and vecr is any v...

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  2. If vecp = (vecbxxvecc)/([vecavecbvecc]), vecq=(veccxxveca)/([veca vecb...

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  3. A( vec a),B( vec b)a n dC( vec c) are the vertices of triangle A B ...

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

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

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

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  9. If veca' = hati + hatj, vecb'= hati - hatj + 2hatk and vecc' = 2hati -...

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

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  11. If the unit vectors veca and vecb are inclined of an angle 2 theta ...

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

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  13. Let veca.vecb=0 where veca and vecb are unit vectors and the vector ve...

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  14. veca and vecb are two given vectors. On these vectors as adjacent side...

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

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

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

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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 na...

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