JEE Advanced 2014 Paper 1Question 1
Let $\vec{a},\vec{b}$ and $\vec{c}$ be three non-coplanar unit vectors such that the angle between every pair of them is $\frac{\pi}{3}$. If $\vec{a}\times\vec{b}+\vec{b}\times\vec{c}=p\vec{a}+q\vec{b}+r\vec{c}$, where $p,q$ and $r$ are scalars, then the value of $\frac{p^2+2q^2+r^2}{q^2}$ is
Vector Algebra
Answer$4$
Solution
Let $V=[\vec a\,\vec b\,\vec c]$. Dot the relation with $\vec a,\vec b,\vec c$ (all dot products $\frac12$): $p+\frac{q+r}2=V$, $\frac{p+r}2+q=0$, $\frac{p+q}2+r=V$. Hence $p=r=V$, $q=-V$ and the ratio is $\frac{V^2+2V^2+V^2}{V^2}=4$.