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Applications of Dot Product

If a⃗ + b⃗ ...

Question

If $\to a + \to b + \to c = 0, | \to a | = 3, ∣ ∣ \to b ∣ ∣ = 5$ and $| \to c | = 7$ , then find the angle between $\to a$ and $\to b$ .

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Solution

$\begin{matrix} \to a + \to b + \to c = 0 \Rightarrow \to a + \to b = - \to c \Rightarrow ∣ ∣ \to a + \to b ∣ ∣ = | - \to c | \Rightarrow {∣ ∣ \to a + \to b ∣ ∣}^{2} = {| \to c |}^{2} \Rightarrow (\to a + \to b) . (\to a + \to b) = 7^{2} \Rightarrow \to a . \to a + 2 \to a . \to b + \to b . \to b = 49 \Rightarrow | \to a |^{2} + 2 \to a . \to b + | \to b |^{2} = 49 \Rightarrow {(3)}^{2} + 2 \to a . \to b + {(5)}^{2} = 49 \Rightarrow \to a . \to b = \frac{15}{2} \end{matrix}$
$\Rightarrow | \to a | | \to b | c o s θ = \frac{15}{2}$ , where $θ$ is the angle between $\to a$ and $\to b$
$\begin{matrix} \Rightarrow 3 \times 5 c o s θ = \frac{15}{2} \Rightarrow c o s θ = \frac{1}{2} \Rightarrow θ = 60^{o} \end{matrix}$
Therefore angle between $\to a$ and $\to b$ is $60^{\circ}$ .

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