Vector Algebra
Scalar triple product; minimization
nta_pyq_2023_jan
Grade 12
Question:
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 $[\vec{u}\,\vec{v}\,\vec{w}]$ is $-\alpha\sqrt{3401}$, and $|\vec{u}\cdot\hat{i}|^2 = \frac{m}{n}$ where m and n are coprime natural numbers, then $m+n$ is equal to _____.
Step-by-Step Solution
Key Concept: Minimize $[\vec{u}\,\vec{v}\,\vec{w}] = \vec{u}\cdot(\vec{v}\times\vec{w})$ over all $\vec{u}$ with $|\vec{u}|=\alpha$: minimum $= -\alpha|\vec{v}\times\vec{w}|$.
$\vec{v}\times\vec{w} = \begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\\alpha&2&-3\\2\alpha&1&-1\end{vmatrix} = \hat{i}(-2+3)-\hat{j}(-\alpha+6\alpha)+\hat{k}(\alpha-4\alpha) = \hat{i}-5\alpha\hat{j}-3\alpha\hat{k}$. $|\vec{v}\times\vec{w}|^2 = 1+25\alpha^2+9\alpha^2=1+34\alpha^2=3401 \Rightarrow \alpha^2=100 \Rightarrow \alpha=10$. $\vec{u}$ minimizes when $\vec{u}\parallel(\vec{v}\times\vec{w})$ direction. $\vec{u} = -10\cdot\frac{(\hat{i}-50\hat{j}-30\hat{k})}{\sqrt{3401}}$. $|\vec{u}\cdot\hat{i}|^2 = \frac{100\cdot1}{3401} = \frac{100}{3401}$. $m+n = 100+3401 = 3501$. Answer: 3501
Correct Answer: 3501