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Question
If \[\vec{a} + \vec{b} + \vec{c} = \vec{0} , \left| \vec{a} \right| = 3, \left| \vec{b} \right| = 5, \left| \vec{c} \right| = 7,\] then the angle between \[\vec{a} \text{ and } \vec{b}\] is
Options
\[\frac{\pi}{6}\]
\[\frac{2\pi}{3}\]
\[\frac{5\pi}{3}\]
\[\frac{\pi}{3}\]
Solution
\[\frac{\pi}{3}\]
\[\text{ Given }, \left| \vec{a} \right| = 3, \left| \vec{b} \right| = 5 \text{ and } \left| \vec{c} \right| = 7 . . . \left( i \right)\]
\[\text{ Let } \theta \text{ be the angle between } \vec{a} \text{ and } \vec{b} .\]
\[\text{ Given that }\]
\[ \vec{a} + \vec{b} + \vec{c} = 0\]
\[ \Rightarrow \vec{a} + \vec{b} = - \vec{c} \]
\[ \Rightarrow \left| \vec{a} + \vec{b} \right| = \left| - \vec{c} \right|^2 \]
\[ \Rightarrow \left| \vec{a} \right|^2 + \left| \vec{b} \right|^2 + 2 \vec{a} . \vec{b} = \left| \vec{c} \right|^2 \]
\[ \Rightarrow 2 \vec{a} . \vec{b} = \left| \vec{c} \right|^2 - \left| \vec{a} \right|^2 - \left| \vec{b} \right|^2 \]
\[ \Rightarrow 2 \vec{a} . \vec{b} = 7^2 - 3^2 - 5^2...................... \left[ \text{ Using } \left( i \right) \right]\]
\[ \Rightarrow 2 \vec{a} . \vec{b} = 15\]
\[ \Rightarrow 2 \left| \vec{a} \right| \left| \vec{b} \right| \cos \theta = 15\]
\[ \Rightarrow 2 \left( 3 \right) \left( 5 \right) \cos \theta = 15 ...................\left[ \text{ Using } \left( i \right) \right]\]
\[ \Rightarrow \cos \theta = \frac{1}{2}\]
\[ \therefore \theta = \frac{\pi}{3}\]
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