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If veca+vecb+vec(c)=vec0, then prove tha...

If `veca+vecb+vec(c)=vec0`, then prove that `vecaxxvecb=vecbxxvec(c)=vec(c)xxveca`.

Answer

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If veca,vecb,vec(c) are the position vectors of the points A,B and C respectively, then prove that the vector vecaxxvecb+vecbxxvec(c)+vec(c)xxveca is perpendicular to the plane of DeltaABC .

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Knowledge Check

  • If vec(a) + 2vec(b) + 4 vec(c )= vec(0) , then vec(a) xx vec(b) + vec(b) xx vec(c )+ vec(c )xx vec(a)=

    A
    0
    B
    `1(vec(b) xx vec(c ))`
    C
    `2(vec(b) xx vec(c ))`
    D
    `7(vec(b) xx vec(c ))`

  • If vec(C) = vec(A) + vec(B) then

    A
    `vec(C)` is always greater than `|vec(A)|`
    B
    C is always equal to A+B
    C
    C is never equal to A+B
    D
    It is possible to have `|vec(C)| lt |vec(A)|` and `|vec(C)| lt |vec(B)|`

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    Explore conceptually related problems

    If vec(a) + 2vec(b) + 4vec(c ) = vec(0) then show that vec(a) xx vec(b) =4 (vec(b) xx vec(c ))= 2(vec(c ) xx vec(a))

    If veca,vecb,vecc are unit vectors such that veca+vecb+vecc=vec0 , find the value of veca*vecb+vecb*vecc+vecc*veca .

    If vec(a) = 7vec(i) -2vec(j) + 3vec(k), vec(b)= 2vec(i) + 8vec(k), vec(c ) =vec(i) + vec(j) + vec(k) , then verify that vec(a) xx (vec(b) + vec(c ))= (vec(a) xx vec(b)) + (vec(a) xx vec(c )) (or) prove that cross product is distributive over addition

    If the vectors vec(a), vec(b), vec(c ) form the sides vec(BC), vec(CA), vec(AB) respectively of triangle ABC then show that vec(a) xx vec(b) = vec(b)xx vec(c ) = vec(c ) xx vec(a) .

    For any three vectors veca,vecb,vec(c) prove that [vecbxxvec(c)" "vec(c)xxveca" "vecaxxvecb]=[vec(a)" "vecb" "vec(c)]^(2)

    If veca=veci-2vecj-3veck,vecb=2veci+vecj-veck and vec(c)=veci+3vecj-2veck , verify that vecaxx(vecbxxvec(c))ne(vecaxxvecb)xxvec(c) .

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