IIT-JEE 2003Question 1
If $\vec u,\vec v,\vec w$ are three non-coplanar unit vectors and $\alpha,\beta,\gamma$ are the angles between $\vec u$ and $\vec v$, $\vec v$ and $\vec w$, $\vec w$ and $\vec u$ respectively and $\vec x,\vec y,\vec z$ are unit vectors along the bisectors of the angles $\alpha,\beta,\gamma$ respectively. Prove that $$\left[\vec x\times\vec y\ \ \vec y\times\vec z\ \ \vec z\times\vec x\right]=\frac1{16}\left[\vec u\ \vec v\ \vec w\right]^2\sec^2\frac\alpha2\sec^2\frac\beta2\sec^2\frac\gamma2.$$
Vector Algebra
Answerproved
Solution
$\vec x=\frac{\vec u+\vec v}{2\cos(\alpha/2)}$, similarly $\vec y,\vec z$. $[\vec x\times\vec y\ \vec y\times\vec z\ \vec z\times\vec x]=[\vec x\,\vec y\,\vec z]^2$ and $[\vec u+\vec v,\vec v+\vec w,\vec w+\vec u]=2[\vec u\,\vec v\,\vec w]$, so $[\vec x\,\vec y\,\vec z]=\frac{2[\vec u\vec v\vec w]}{8\cos\frac\alpha2\cos\frac\beta2\cos\frac\gamma2}$. Squaring gives $\frac1{16}[\vec u\vec v\vec w]^2\sec^2\frac\alpha2\sec^2\frac\beta2\sec^2\frac\gamma2$.