InterviewSolution
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InterviewSolution
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If A(veca).B(vecb) and C(vecc) are three non-collinear point and origin does not lie in the plane of the points A, B and C, then for any point P(vecP) in the plane of the triangleABC such that vector vec(OP) is botto plane of trianglABC, show that vec(OP)=([vecavecbvecc] (vecaxxvecb+vecbxxvecc+veccxxveca))/(4Delta^(2)) |
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Answer» Solution :P LIES in the plane of A,B and C , therefore, `vec(AP).(vec(Bp) xx vec(CP))=0` `Rightarrow (vecp.veca).(veccxxvecp+vecpxxvecb+vecbxxvecc)=0` `or 0 +0+vecp . (vecbxxvecc)-veca.(veccxxvecp)` `-veca.(vecpxxvecb)-veca.(vecbxxvecc)=0` `or vecp . (vecbxxvecc)=vecp.(veccxxveca) +vecp.(vecaxxvecb)` `-veca . (VECB xx vecc) =0` `or vecp.(vecbxxvecc+veccxxveca+vecaxxvecb) = [veca vecbvecc]` now vector perependicular to the plane ABC is `(vecb xx vecc +vecc xx veca +veca xxvecb)` Let ` vec(OP) = LAMBDA(veca xx vecb + vecb +vecb xx vecc +vecc xx veca)` since `vec(OP).(vecaxxvecb+vecbxxvecc+veccxxveca)=[veca vecbvecc]` `or ([veca vecb vecc])/(4Delta^(2))` `vec(OP) = ([veca vecb vecc] (veca xxvecb + vecb xxvecc +veccxxveca))/(4Delta^(2))` |
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