Let ABCDEF be a regular hexagon. If $\vec{A D} = m \vec{B C} a n d \vec{C F} = n \vec{A B}$ then what is mn equal to

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Question

This question was previously asked in

NDA-II 2024 (Maths) Official Paper (Held On: 01 Sept, 2024)

Answer 1

Calculation:

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$\vec{A B} = 2 \vec{F C}$

⇒ $\vec{A B} = - 2 \vec{C F}$

⇒ $n = - \frac{1}{2}$

Also,

$\vec{A D} = 2 \vec{B C} \Rightarrow m = 2$

Now,

$m n = 2 (- \frac{1}{2}) = - 1$

∴ The final value of is -1.

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