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Question
Find the angle between two vectors \[\vec{a} \text{ and } \vec{b}\]
\[\left| \vec{a} \right| = 3, \left| \vec{b} \right| = 3 \text{ and } \vec{a} \cdot \vec{b} = 1\]
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Solution
\[\ \text{ Let } \theta \text{ be the angle between } \vec{a} \text{ and } \vec{b} .\]
\[\text{ Given that }\]
\[\left| \vec{a} \right| = 3, \left| \vec{b} \right| = 3 \text{ and } \vec{a} . \vec{b} = 1 . . . \left( 1 \right)\]
\[\text{ We know that }\]
\[ \vec{a} . \vec{b} = \left| \vec{a} \right| \left| \vec{b} \right| \cos \theta\]
\[ \Rightarrow 1 = \left( 3 \right)\left( 3 \right) \cos \theta \left[ \text{ Using } \left( 1 \right) \right]\]
\[ \Rightarrow \cos \theta = \frac{1}{\left( 3 \right)\left( 3 \right)} = \frac{1}{9}\]
\[ \Rightarrow \theta = \cos^{- 1} \left( \frac{1}{9} \right)\]
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