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Machiavelli

If |A⃗|= 2 an...

Question

If $| \to A | = 2$ and $| \to B | = 4$ , then match the relations in column $I$ with the angle between $\to A \to B$ and in column $I I$ .

$\begin{matrix} Column I & Column II (a) | \to A \times \to B | = 0 & (i) θ = 30^{\circ} (b) | \to A \times \to B | = 8 & (i i) θ = 45^{\circ} (c) | \to A \times \to B | = 4 & (i i i) θ = 90^{\circ} (d) | \to A \times \to B | = 4 \sqrt{2} & (i v) θ = 0^{\circ} \end{matrix}$

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Solution

Formula used: $\to A \times \to B = | \to A | | \to B | sin θ$

Solution:

$(a) | \to A \times \to B | = 0$
$\to A \times \to B = | \to A | | \to B | sin θ$
$sin θ = 0$
$θ = 0^{\circ}$

$(b) | \to A \times \to B | = 8$
$| \to A \times \to B | = | \to A | | \to B | sin θ$
$8 = 2 \times 4 \times sin θ$
$sin θ = 1$
$θ = 90^{\circ}$

$(c) | \to A \times \to B | = 4$
$| \to A \times \to B | = | \to A | | \to B | sin θ$
$4 = 2 \times 4 \times sin θ$
$sin θ = \frac{1}{2}$
$θ = 30^{\circ}$

$(d) | \to A \times \to B | = 4 \sqrt{2}$
$| \to A \times \to B | = | \to A | | \to B | sin θ$
$4 \sqrt{2} = 2 \times 4 \times sin θ$
$sin θ = \frac{1}{\sqrt{2}}$
$θ = 45^{\circ}$

Final answer:
$(a)$ matches with $(i v)$
$(b)$ matches with $(i i i)$
$(c)$ matches with $(i)$
$(d)$ matches with $(i i)$

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