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Applications of Dot Product

If a and ...

Question

If $\to a$ and $\to b$ are unit vectors inclined at angle of $30^{o}$ to each other, then which one of the following is correct?

∣ ∣ ∣ \to a + \to b ∣ ∣ ∣ = 2

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∣ ∣ ∣ \to a + \to b ∣ ∣ ∣ > 2

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∣ ∣ ∣ \to a + \to b ∣ ∣ ∣ > 1

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1 < ∣ ∣ ∣ \to a + \to b ∣ ∣ ∣ < 2

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Solution

The correct option is D $1 < ∣ ∣ ∣ \to a + \to b ∣ ∣ ∣ < 2$
$| \to a | = | \to b | = 1$
$θ = 30^{o}$
$| \to a + \to b |^{2} = | \to a |^{2} + | \to b |^{2} + 2 \to a, \to b$
$\Rightarrow | \to a + \to b |^{2} = 1 + 1 + 2 | \to a | | \to b | cos 30^{o}$
$= 2 + 2 \times \frac{\sqrt{3}}{2}$

$\Rightarrow | \to a + \to b |^{2} = 2 + \sqrt{3}$
$| \to a + \to b | = \sqrt{2 + \sqrt{3}}$
$1 < | \to a + \to b | < 2$

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