The magnitudes of four pairs of displacement vectors are given. Which pairs of displacement vectors cannot be added to give a resultant vector of magnitude 4 $c m$ ?

2 c m, 3 c m

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1 c m, 3 c m

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1 c m, 5 c m

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1 c m, 7 c m

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Solution

The correct option is D $1 c m, 7 c m$
$\begin{matrix} T h e m a g n i t u d e R o f t h e r e s u l tan t o f t w o v e c t o r s A a n d B d e p e n d s u p o n t h e m a g n i t u d e s o f A a n d B a n d t h e a n g l e θ b e t w e e n t h e m a n d i s g i v e n b y R^{2} = A^{2} + B^{2} + 2 A B cos θ w h e n θ = 0, R i s max i m u m g i v e n b y {R^{2}}_{max} = A + B w h e n θ = 180^{\circ}, R i s m i n i m u m g i v e n b y R_{min}^{2} = A^{2} + B^{2} - 2 A B o r, R_{min} = A - B T h u s t h e m a g n i t u d e o f r e s u l tan t w i l l l i e b e t w e e n A - B a n d A + B, . \end{matrix}$
$\begin{matrix} N o w, O n c h e c k o p t i o n (D) | A - B | = | 1 - 7 | = 6 | A + B | = | 1 + 7 | = 8 S o, 6 \leq R \leq 8 ∴ R \neq 4 H e n c e, t h e o p t i o n D i s t h e c o r r e c t a n s w e r . \end{matrix}$

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The magnitudes of four pairs of displacement vectors are given. Which pairs of displacement vectors cannot be added to give a resultant vector of magnitude 4cm ?

The magnitudes of four pairs of displacement vectors are given. Which pairs of displacement vectors cannot be added to give a resultant vector of magnitude 4 $c m$ ?