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If $\left| {\vec A } \right|\, = \,2$ and $\left| {\vec B } \right|\, = \,4$ then match the relation in Column $-I$ with the angle $\theta $ between $\vec A$ and $\vec B$ in Column $-II$.
| Column $-I$ | Column $-II$ |
| $(a)$ $\left| {\vec A \, \times \,\,\vec B } \right|\, = \,\,0$ | $(i)$ $\theta = \,{30^o}$ |
| $(b)$ $\left| {\vec A \, \times \,\,\vec B } \right|\, = \,\,8$ | $(ii)$ $\theta = \,{45^o}$ |
| $(c)$ $\left| {\vec A \, \times \,\,\vec B } \right|\, = \,\,4$ | $(iii)$ $\theta = \,{90^o}$ |
| $(d)$ $\left| {\vec A \, \times \,\,\vec B } \right|\, = \,\,4\sqrt 2$ | $(iv)$ $\theta = \,{0^o}$ |
Solution
Given $|\mathrm{A}|=2$ and $|\mathrm{B}|=4$
$(a)$ $|\vec{A} \times \vec{B}|=A B \sin \theta=0$
$\therefore \sin \theta=0$
$\therefore \theta=0$
$\therefore$ Option $(a)$ matches with option $(iv)$.
$(b)$ $|\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}|=\mathrm{AB} \sin \theta=8$ $\therefore 2 \times 4 \sin \theta=8$ $\therefore \sin \theta=1=\sin 90^{\circ}$ $\therefore \theta=90^{\circ}$ $\therefore$ Option (b) matches with option (iii)
$\therefore$ Option (b) matches with (c) $|\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}|=\mathrm{AB} \sin \theta=4$
$(c)$ $|\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}|=\mathrm{AB}$ $\therefore 2 \times 4 \sin \theta=4$
$\therefore \sin \theta=\frac{1}{2}=\sin 30^{\circ}$
$\therefore \theta=30^{\circ}$
$\therefore$ Option $(c)$ matches with option $(i)$.
$(d)$ $|\vec{A} \times \vec{B}|=A B \sin \theta=4 \sqrt{2}$
$\therefore 2 \times 4 \sin \theta=4 \sqrt{2}$
$\therefore \sin \theta=\frac{1}{\sqrt{2}}=\sin 45^{\circ}$
$\therefore \theta=45^{\circ}$
Option $(d)$ matches with option $(ii)$.
