If `A( vec a),B( vec b)a n dC( vec c)` are three non-collinear points and origin does not lie in the plane of the points `A ,Ba n dC ,` then point `P( vec p)` in the plane of the ` A B C` such that vector ` vec O P` is `_|_` to planeof ` A B C` , show that ` vec O P=([ vec a vec b vec c]( vec axx vec b+ vec bxx vec c+ vec cxx vec a))/(4^2),w h e r e` is the area of the ` A B Cdot`

Explanation

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))`