`A (veca), B (vecb) and C (vecc)` are the vertices of triangle ABC and `R(vecr)` is any point in the plane of triangle ABC, then `vecr, (veca xx vecb + vecb xx vecc + vecc xx veca ) ` is always equal to

Options

A. zero
B. ` [veca vecb vecc ] `
C. `- [ veca vecb vecc]`
D. none of these

Answer

b

Explanation

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]`