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If the volume of the parallelopiped form...

If the volume of the parallelopiped formed by the vectors `veca, vecb, vecc` as three coterminous edges is 27 units, then the volume of the parallelopiped having `vec(alpha)=veca+2vecb-vecc, vec(beta)=veca-vecb`
and `vec(gamma)=veca-vecb-vecc` as three coterminous edges, is

A

27

B

9

C

81

D

none of these

Text Solution

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The correct Answer is:
C
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The volume of a parallelopiped whose coterminous edges are 2veca , 2vecb , 2 vec c , is

If V is the volume of the parallelopiped having three coterminous edges as veca,vecb and vecc , then the volume of the parallelopiped having three coterminous edges as vec(alpha)=(veca.veca)veca+(veca.vecb)vecb+(veca.vecc)vecc vec(beta)=(veca.vecb)veca+(vecb.vecb)vecb+(vecb.vecc)vecc vec(gamma)=(veca.vecc)veca+(vecb.vecc)vecb+(vecc.vecc)vecc is

The volume of a tetrahedron fomed by the coterminus edges veca , vecb and vecc is 3 . Then the volume of the parallelepiped formed by the coterminus edges veca +vecb, vecb+vecc and vecc + veca is

Statement 1: If V is the volume of a parallelopiped having three coterminous edges as veca, vecb , and vecc , then the volume of the parallelopiped having three coterminous edges as vec(alpha)=(veca.veca)veca+(veca.vecb)vecb+(veca.vecc)vecc vec(beta)=(veca.vecb)veca+(vecb.vecb)vecb+(vecb.vecc)vecc vec(gamma)=(veca.vecc)veca+(vecb.vecc)vecb+(vecc.vecc)vecc is V^(3) Statement 2: For any three vectors veca, vecb, vecc |(veca.veca, veca.vecb, veca.vecc),(vecb.veca,vecb.vecb,vecb.vecc),(vecc.veca,vecc.vecb,vecc.vecc)|=[(veca,vecb, vecc)]^(3)

If |veca|=5, |vecb|=3, |vecc|=4 and veca is perpendicular to vecb and vecc such that angle between vecb and vecc is (5pi)/6 , then the volume of the parallelopiped having veca, vecb and vecc as three coterminous edges is

If V is the volume of the parallelepiped having three coterminous edges as veca,vecb and vecc , then the volume of the parallelepiped having three coterminous edges as vecalpha = (veca.veca)veca+(veca.vecb)vecb+(veca.vecc)vecc , vecbeta=(vecb.veca)veca+(vecb.vecb)+(vecb.vecc)vecc and veclambda=(vecc.veca)veca+(vecc.vecb)vecb+(vecc.vecc)vecc is

Statement 1: Let V_(1) be the volume of a parallelopiped ABCDEF having veca, vecb, vecc as three coterminous edges and V_(2) be the volume of the parallelopiped PQRSTU having three coterminous edges as vectors whose magnitudes are equal to the areas of three adjacent faces of the parallelopiped ABCDEF . Then V_(2)=2V_(1)^(2) Statement 2: For any three vectors vec(alpha), vec(beta), vec(gamma) [vec(alpha)xxvec(beta),vec(beta)xxvec(gamma),vec(gamma)xxvec(alpha)]=[(vec(alpha),vec(beta),vec(gamma))]^(2)

If veca, vecb, vecc are vectors such that veca.vecb=0 and veca + vecb = vecc then:

If [(veca,vecb,vecc)]=3 , then the volume (in cubic units) of the parallelopiped with 2veca+vecb,2vecb+vecc and 2vecc+veca as coterminous edges is

ARIHANT MATHS-PRODUCT OF VECTORS-Exercise (Questions Asked In Previous 13 Years Exam)
  1. If the volume of the parallelopiped formed by the vectors veca, vecb, ...

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  2. Let O be the origin and let PQR be an arbitrary triangle. The point S ...

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  3. Let O be the origin and OX, OY, OZ be three unit vectors in the direct...

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  4. Let O be the origin, and O X , O Y , O Z be three unit vectors ...

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  5. Let a, b and c be three unit vectors such that atimes(btimesc)=(sqrt(3...

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  6. Let vec(a) , vec(b) and vec(c) be three non-zero vectors such that no ...

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  7. If a, b and c are unit vectors satisfying |a-b|^(2)+|b-c|^(2)+|c-a|^(2...

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  8. The vector(s) which is/are coplanar with vectors hat(i)+hat(j)+2hat(k)...

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  9. Let vec a =hat i+hat j+hat k,vec b=hat i -hat j+hat k and vec c =hat ...

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  10. Two adjacent sides of a parallelogram ABCD are given by vec(AB)=2hati+...

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  11. Let P, Q, R and S be the points on the plane with position vectors -2h...

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  12. If a and b are vectors in space given by a=(hat(i)-2hat(j))/(sqrt(5)) ...

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  13. If veca,vecb,vecc and vecd are unit vectors such that (vecaxxvecb)*(...

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  14. The edges of a parallelopiped are of unit length and are parallel to ...

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  15. Lelt two non collinear unit vectors hata and hatb form and acute angle...

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  16. Let the vectors PQ, QR, RS, ST, TU and UP represent the sides of a reg...

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  17. The number of distinct real values of lambda, for which the vectors -l...

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  18. Let veca,vecb,vecc be unit vectors such that veca+vecb+vecc=vec0. Whic...

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  19. Let A be vector parallel to line of intersection of planes P1 and P2. ...

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  20. Let a=hat(i)+2hat(j)+hat(k), b=hat(i)-hat(j)+hat(k), c=hat(i)+hat(j)-h...

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  21. If vec a , vec b and vec c are three non-zero, non coplanar vecto...

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