Let `vec a, vec b, and vec c` be three non coplanar unit vectors such that the angle between every pair of them is `pi/3`. If `vec a xx vec b+ vecb xx vec x=p vec a + q vec b + r vec c` where p,q,r are scalars then the value of `(p^2+2q^2+r^2)/(q^2)` is

Options

A. `2`
B. `4`
C. `6`
D. `8`

Answer

B

Explanation

We have
`vecaxxvecb+vecbxxvec=pveca+qvecb+rvecc`
Taking dot product with `veca` we obtain
`veca.(vecaxxvecb)+veca.(vecbxxvecc)=p(veca.veca)+q(veca.vecb)+r(veca.vecc)`
`=[(veca, vecb, vecc)]=p+q/2+r/2`…………..ii
Taking dot product on both sides of (i) by `vecb`, we get
`vecb.(vecaxxvecb)+vecb.(vecbxxvecc)-p(vecb.veca)+q(vecb.vecb)+r(vecc.vecc)`
`implies0=p/2+q+r/2`................iii
Taking dot product on both sides of (i) by `vecc` we ger
`vecc.(vecaxxvecb)+vecc.(vecbxxvecc)=p(vecc.veca)+q(vecc.vecb)+r(vecc.vecc)`
`implies [(vecc, veca, vecb)]=p/2+q/2+r`
`implies[(veca, vecb, vecc)]=p/2+q/2+r`..........iv
From ii and iv we get
`p+q/2+r/2=p/2+q/2+rimpliesp=r`
Putting `p=r` in (iii) we get `q=-r`Thus, we obtain
`p=r=-q`
`:.(p^(2)+2q^(2)+r^(2))/(q^(2))=((-q)^(2)+2q^(2)+(-q)^(2))/(q^(2))=r`