Let v¯=αi^+2j^−3k^, w¯=2αi^+j^−k^ and u¯ be a vector such that |u^|=a>0 . If the minimum value of the scalar triple product [u¯ v¯ w¯] is −α3401 and |u¯⋅i^|2=mn , where m and n are coprime natural numbers, then is equal to ________

We have v=αi^+2j^−3k^,w=2αi^+j^−k^|u^|=αNow, [u v w=u.(v×w)]=|u||v×w|cosθ=α|v×w|cosθFor minimum value of [u v w],θ=π∴[u v w]min=−α[v×w] ⇒|v×w|=3401 Now, v×w=| i j kα 2 −32α 1 −1| =i−5αj−3αk Now, |v×w|=3401 ⇒α=10 [∵α>0] ∴v×w=i−50j−30k Again, u=10(−v×w|v×w|) =−103401(i−50j−30k)Now, |u.i|2=(−103401)2=1003401 ∴m=100,n=3401 . So, m+n=3501

AI Mentor

Check Your IQ

Free Expert Demo

Try Test

Courses

Dropper NEET Course Dropper JEE Course Class - 12 NEET Course Class - 12 JEE Course Class - 11 NEET Course Class - 11 JEE Course Class - 10 Foundation NEET Course Class - 10 Foundation JEE Course Class - 10 CBSE Course Class - 9 Foundation NEET Course Class - 9 Foundation JEE Course Class -9 CBSE Course Class - 8 CBSE Course Class - 7 CBSE Course Class - 6 CBSE Course

Offline Centres

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

see full answer

High-Paying Jobs That Even AI Can’t Replace — Through JEE/NEET

🎯 Hear from the experts why preparing for JEE/NEET today sets you up for future-proof, high-income careers tomorrow.

An Intiative by Sri Chaitanya

answer is 3501.

(Unlock A.I Detailed Solution for FREE)

Best Courses for You

JEE

NEET

Foundation JEE

Foundation NEET

CBSE

Detailed Solution

We have $v = α \hat{i} + 2 \hat{j} - 3 \hat{k}, w = 2 α \hat{i} + \hat{j} - \hat{k} | \hat{u} | = α$
Now, $[u v w = u . (v \times w)] = | u | | v \times w | \cos θ = α | v \times w | \cos θ$
For minimum value of $[u v w], θ = π$
$∴ {[u v w]}_{\min} = - α [v \times w]$    $\Rightarrow | v \times w | = \sqrt{3401}$
Now, $v \times w = | \begin{array}{l} i j k \\ α 2 - 3 \\ 2 α 1 - 1 \end{array} |$ $= i - 5 α j - 3 α k$
Now,    $| v \times w | = \sqrt{3401} \Rightarrow α = 10 [∵ α > 0] ∴ v \times w = i - 50 j - 30 k$

Again,    $u = 10 (- \frac{v \times w}{| v \times w |}) = \frac{- 10}{\sqrt{3401}} (i - 50 j - 30 k)$
Now, ${| u . i |}^{2} = {(\frac{- 10}{\sqrt{3401}})}^{2} = \frac{100}{3401}$    $∴ m = 100, n = 3401$
. So,    $m + n = 3501$