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 $-II$
$(a)$ $\left| {\vec A \, \times \,\,\vec B } \right|\, = \,\,0$	$(i)$ $\theta = \,{30^o}$
$(b)$ $\left| {\vec A \, \times \,\,\vec B } \right|\, = \,\,8$	$(ii)$ $\theta = \,{45^o}$
$(c)$ $\left| {\vec A \, \times \,\,\vec B } \right|\, = \,\,4$	$(iii)$ $\theta = \,{90^o}$
$(d)$ $\left| {\vec A \, \times \,\,\vec B } \right|\, = \,\,4\sqrt 2$	$(iv)$ $\theta = \,{0^o}$

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 $(ii)$.