$If | A | = 2 and | B | = 4, then match the relations in Column I with the angle θ between A and B in Column II .$
Column I Column II
(p) $| \vec{A} \times \vec{B} | = 0$ (i) $θ = 30^{0}$
(q) $| \vec{A} \times \vec{B} | = 8$ (ii) $θ = 45^{0}$
r $| \vec{A} \times \vec{B} | = 4$ (iii) $θ = 90^{0}$
(s) $| \vec{A} \times \vec{B} | = 4 \sqrt{2}$ (iv) $θ = 0^{0}$
Mark the correct choice from the given codes.
Codes

Moderate

Solution

Given $| A | = 2 and | B | = 4$
(p) $| \vec{A} \times \vec{B} | = ABsinθ = 0$
$\Rightarrow 2 \times 4 \times sinθ = 0$
$\Rightarrow sinθ = 0 = \sin 0^{0} \Rightarrow θ = 90^{0}$
Hence, option (p) matches with option (iv).
(q) $| \vec{A} \times \vec{B} | = ABsinθ = 8$
$\Rightarrow 2 \times 4 sinθ = 8$
$\Rightarrow sinθ = 1 = \sin 90^{0} \Rightarrow θ = 90^{0}$
Hence, option (q) matches with option(iii)
r. $| \vec{A} \times \vec{B} | = ABsinθ = 4 \Rightarrow 2 \times 4 sinθ = 4$
$sinθ = \frac{1}{2} = \sin 30^{0} \Rightarrow θ = 30^{0}$
Hence, option r matches with option (i)
(s). $| \vec{A} \times \vec{B} | = ABsinθ = 4 \sqrt{2}$
$\Rightarrow 2 \times 4 sinθ = 4 \sqrt{2}$
$\Rightarrow sinθ = \frac{1}{\sqrt{2}} = \sin 45^{0} \Rightarrow θ = 45^{0}$
Hence, option(s) matches with option (ii)

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