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Let vec a, vec b, and vec c be three non...

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

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The correct Answer is:
`(4)`
`|veca|=|vecb|=|vecc|=1`
` veca.vecb=vecb. vecc=vecc.veca=1//2`
Also ` vecaxxvecb+vecbxxvecc=pveca+qvecb+rvecc`
`implies veca.(vecbxxvecc) = p +q(veca.vecb) +r(veca.vecc)`
`implies veca.(vecbxxvecc ) = p+q(veca.vecb) r(veca.vecc)`
`therefore p+(q)/(2)+(r)/(2)=[vecavecbvecc]`
similarly taking dot product with vector `vecb` , we get
`(p)/(2)+q+(r)/(2)=0`
And, taking dot product with vector `vecc`, we get
`(p)/(2)+(q)/(2)+r=[veca vecbvecc]`
solving,(1),(2) and (3) , we get
` p=r=-q`
`implies (p^(2)+2q^(2)+r^(2))/(q^(2))=4`
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