JEE Main 2023 (Online) 1st February Morning ShiftQuestion 1
Let $\vec v=\alpha\hat i+2\hat j-3\hat k$, $\vec w=2\alpha\hat i+\hat j-\hat k$, and $\vec u$ be a vector such that $|\vec u|=\alpha>0$. If the minimum value of the scalar triple product $\left[\vec u\ \vec v\ \vec w\right]$ is $-\alpha\sqrt{3401}$, and $\left(\vec u\cdot\hat i\right)^2=\frac mn$ where $m$ and $n$ are coprime natural numbers, then $m+n$ is equal to ______.
Vector Algebra
Answer$3501$
Solution
Min of $\vec u\cdot(\vec v\times\vec w)=-\alpha|\vec v\times\vec w|$, $\vec v\times\vec w=(1,-5\alpha,-3\alpha)$, so $\alpha\sqrt{1+34\alpha^2}=\alpha\sqrt{3401}$ gives $\alpha=10$. Then $\vec u$ is antiparallel to $(1,-50,-30)$: $(\vec u\cdot\hat i)^2=\frac{100}{3401}$. So $m+n=3501$.