Consider three vectors a , b , c . Let | a |=2,| b |=3 and a = b × c . If α ∈ [0, (π)/(3)] is the angle between the vectors b and c , then the minimum value of 27| c - a |^2 is equal to:

Correct answer: B. 124 aligned & | c - a |=| c |^2+| a |^2-2 a · c & =| c |^2+4-0 & a = b × c & | a |=| b × c | & 2=3| c | α & | c |=(2)/(3) cosec α α ∈[0, (π)/(3)] & | c |_ =(2)/(3) × 2 3 cosec α ∈[ 2 3 , ∞) & 27| c - a |_ ^2=27((16)/(27)+4)=124 aligned

JEE Main 20245 AprilEvening Shift

Exam: JEE Main
Year: 2024
Date: 5 April
Shift: Evening Shift
Subject: Mathematics
Chapter: Vector Algebra

JEE Main 2024 · 5 April · Evening Shift · Mathematics · Vector Algebra

Consider three vectors a , b , c . Let | a |=2,| b |=3 and a = b ×...

Consider three vectors

\vec{a}, \vec{b}, \vec{c}

. Let

|\vec{a}|=2,|\vec{b}|=3 and

\vec{a}=\vec{b} \times \vec{c}

. If

\alpha \in

\left[0, \frac{\pi}{3}\right]

is the angle between the vectors

\vec{b}

and

\vec{c}

, then the minimum value of

27|\vec{c}-\vec{a}|^2

is equal to:

Options

A110
B124
C121
D105

Correct answer

B. 124

Step-by-step solution

\begin{aligned} & |\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}|=|\overrightarrow{\mathrm{c}}|^2+|\overrightarrow{\mathrm{a}}|^2-2 \overline{\mathrm{a}} \cdot \overline{\mathrm{c}} \\ & =|\overrightarrow{\mathrm{c}}|^2+4-0 \\ & \because \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}} \\ & |\overrightarrow{\mathrm{a}}|=|\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}| \\ & 2=3|\overrightarrow{\mathrm{c}}| \sin \alpha \\ & |\overrightarrow{\mathrm{c}}|=\frac{2}{3} \operatorname{cosec} \alpha \quad \alpha \in\left[0, \frac{\pi}{3}\right] \\ & |\overrightarrow{\mathrm{c}}|_{\min }=\frac{2}{3} \times \frac{2}{\sqrt{3}} \quad \operatorname{cosec} \alpha \in\left[\frac{2}{\sqrt{3}}, \infty\right) \\ & \Rightarrow 27|\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}|_{\min }^2=27\left(\frac{16}{27}+4\right)=124\end{aligned}

Practice Vector Algebra on Quantrex Academy →

More from Vector Algebra