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If three vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ satisfy $\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0}$ and $|\mathbf{a}|=3,|\mathbf{b}|=5,|\mathbf{c}|=7$, then the angle between $\mathbf{a}$ and $\mathbf{b}$ is
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Verified Answer
The correct answer is:
$60^{\circ}$
Given, $\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0} \Rightarrow \mathbf{a}+\mathbf{b}=-\mathbf{c}$
Squaring on both sides,
$\begin{aligned}
& \Rightarrow|\mathbf{a}|^2+|\mathbf{b}|^2+2|\mathbf{a}||\mathbf{b}| \cos \theta=|-\mathbf{c}|^2 \\
& \Rightarrow 9+25+30 \cos \theta=49 \Rightarrow \cos \theta=\frac{1}{2} \Rightarrow \theta=60^{\circ} .
\end{aligned}$
Squaring on both sides,
$\begin{aligned}
& \Rightarrow|\mathbf{a}|^2+|\mathbf{b}|^2+2|\mathbf{a}||\mathbf{b}| \cos \theta=|-\mathbf{c}|^2 \\
& \Rightarrow 9+25+30 \cos \theta=49 \Rightarrow \cos \theta=\frac{1}{2} \Rightarrow \theta=60^{\circ} .
\end{aligned}$
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