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Question
What is the angle between vectors \[\vec{a} \text{ and } \vec{b}\] with magnitudes 2 and \[\sqrt{3}\] respectively? Given \[\vec{a} . \vec{b} = \sqrt{3} .\]
Solution
\[\text{ Let } \theta \text{ be the angle between } \vec{a} \text{ and } \vec{b} .\]
\[\text{ Given that }\]
\[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = \sqrt{3} \text{ and } \vec{a} . \vec{b} = \sqrt{3}\]
\[\text{ We know that }\]
\[ \vec{a} . \vec{b} = \left| \vec{a} \right| \left| \vec{b} \right| \cos \theta\]
\[ \Rightarrow \sqrt{3} = \left( 2 \right)\left( \sqrt{3} \right) \cos \theta\]
\[ \Rightarrow \cos \theta = \frac{\sqrt{3}}{2\sqrt{3}}\]
\[ \Rightarrow \cos \theta = \frac{1}{2}\]
\[ \Rightarrow \theta = \cos^{- 1} \left( \frac{1}{2} \right) = \frac{\pi}{3}\]
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