Let v = i ^ + 2 j ^ - 3 k ^ , w = 2 i ^ + j ^ - k ^ and u be a vector such that | u | = 0 . If the minimum value of the scalar triple product [ u v w ] is - 3401 , and | u · i ^ | 2 = m n where m and n are coprime natural numbers, then

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Solution

Q.
Let $\vec{v} = α \hat{i} + 2 \hat{j} - 3 \hat{k},$ $\vec{w} = 2 α \hat{i} + \hat{j} - \hat{k}$ and $\vec{u}$ be a vector such that $| \vec{u} | = α > 0 .$ If the minimum value of the scalar triple product $[\vec{u} \vec{v} \vec{w}] is - α \sqrt{3401},$ and $| \vec{u} \cdot \hat{i} |^{2}$ $= \frac{m}{n}$ where $m$ and $n$ are coprime natural numbers, then $m + n$ is equal to _________ . [2023]

Ans.
(3501)

$\vec{v} = α \hat{i} + 2 \hat{j} - 3 \hat{k},$ $\vec{w} = 2 α \hat{i} + \hat{j} - \hat{k}$ and $| \vec{u} |$ $= α$

$\vec{v} \times \vec{w} =$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ α & 2 & - 3 \\ 2 α & 1 & - 1 \end{matrix} |$

$= \hat{i} (- 2 + 3) - \hat{j} (- α + 6 α) + \hat{k} (α - 4 α) = \hat{i} - 5 α \hat{j} - 3 α \hat{k}$

$Now, [\vec{u} \vec{v} \vec{w}] = \vec{u} \cdot (\vec{v} \times \vec{w}) = | \vec{u} |$ $| \hat{i} - 5 α \hat{j} - 3 α \hat{k} |$ $\cos θ$

$= α \sqrt{1 + 25 α^{2} + 9 α^{2}} \cos θ = α \sqrt{1 + 34 α^{2}} \cos θ$

$Minimum value of scalar triple product is - α \sqrt{1 + 34 α^{2}}$ $(∵ α > 0, - 1 \leq \cos θ \leq 1)$

$\Rightarrow - α \sqrt{1 + 34 α^{2}} = - α \sqrt{3401}$

$\Rightarrow 1 + 34 α^{2} = 3401$

$\Rightarrow α^{2} = 100$

$\Rightarrow α = 10$ $[∵ α > 0]$

$\vec{v} \times \vec{w} = \hat{i} - 50 \hat{j} - 30 \hat{k}$

$\vec{u} = \frac{- 10 (\hat{i} - 50 \hat{j} - 30 \hat{k})}{\sqrt{3401}}$ $\Rightarrow$ $| \vec{u} \cdot \hat{i} |^{2}$ $= \frac{100}{3401} = \frac{m}{n}$

$\Rightarrow m + n = 3501$

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