Let `hata,vecb and vecc` be the non-coplanar unit vectors. The angle between `hatb and hatc is alpha "between" hatc and hata is beta and "between" hata and hatb is gamma`. If `A(hatacos alpha),B(hatbcosbeta) and C(hatc cosgamma),` then show that in triangle ABC, `(|hataxx(hatbxxhatca)|)/(sinA)=(|hatbxx(hatcxxhata)|)/sinB = (|hatcxx(hataxxhatb)|)/sinC=(prod|hataxx(hat xx hatc|))/(sumsin alpha-cosbeta. cos gammahatn_(1))` where `hatn_(1)=(hatbxxhatc)/(|hatbxxhatc|),hatn_(2)=(hatcxxhata)/(|hatcxxhata|)and hatn_(3)=(hataxxhatb)/(|hataxxhatb|)`

Explanation

Since `veca,vecb and vecc` are non - coplanar, vectors `vecaxxvecb,vecb xxveccandveccxxveca` are also non-coplanar. Let
`vecd=l(vecbxxvecc)+vecm(veccxxveca)+vecn(vecaxxvecb)`
now multiplying both sides of (i) scalarly by `veca` we have
`veca.vecd=lveca.(vecbxxvecc)+mveca.(veccxxveca)+nveca.(vecaxxvecb)=l[vecavecc veca]([veca vecc veca]=0=[veca veca vecb])`
`l=(veca.vecd)//[veca vecb vecc]`
putting these values oif l,nm and n and (i) , we get the required relation.