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Question
If \[\vec{a} \text{ and } \vec{b}\] are two vectors such that \[\left| \vec{a} \right| = 4, \left| \vec{b} \right| = 3 \text{ and } \vec{a} \cdot \vec{b} = 6\] find the angle between \[\vec{a} \text{ and } \vec{b} .\]
Solution
\[ \text{ Let } θ\text{ be the angle between } \vec{a} \text{ and } \vec{b} . \]
Given that
\[ \vec{a} . \vec{b} = 6\]
\[ \Rightarrow \left| \vec{a} \right| \left| \vec{b} \right| \cos \theta = 6\]
\[ \Rightarrow \left( 4 \right)\left( 3 \right) \cos \theta = 6\]
\[ \Rightarrow 12 \cos \theta = 6\]
\[ \Rightarrow \cos \theta = \frac{6}{12} = \frac{1}{2}\]
\[ \Rightarrow \theta = \cos^{- 1} \left( \frac{1}{2} \right) = \frac{\pi}{3}\]
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