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 ____.

JEE Main · 2023 · 01 Feb 2023 · Shift 1 · Vector Algebra

Let

v = α \hat{i} + 2 \hat{j} - 3 \hat{k}

w = 2 α \hat{i} + \hat{j} - \hat{k}

, and

u

be a vector such that

∣ u ∣ = α > 0

. If the minimum value of the scalar triple product

[u v w]

- α 3401

, and

∣ u \cdot \hat{i} ∣^{2} = \frac{m}{n}

where

m

and

n

are coprime natural numbers, then

m + n

is equal to ____.

Answer: 3501

The triple product

[u v w] = u \cdot (v \times w)

is least when

u

is antiparallel to

v \times w

, giving

- ∣ u ∣ ∣ v \times w ∣ = - α 1 + 34 α^{2}

, since

v \times w = \hat{i} - 5 α \hat{j} - 3 α \hat{k}

. Equating

1 + 34 α^{2} = 3401

gives

α^{2} = 100

, so

α = 10

and

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

with

∣ v \times w ∣^{2} = 3401

. Then

u = - 10 \frac{v \times w}{∣ v \times w ∣}

, so

∣ u \cdot \hat{i} ∣^{2} = \frac{100}{3401}

, and with

m = 100, n = 3401

the value of

m + n

3501

JEE Main · 2023 · 01 Feb 2023 · Shift 1 · Integer answer