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Question
Vectors \[\vec{a} \text{ and } \vec{b}\] \[\left| \vec{a} \right| = \sqrt{3}, \left| \vec{b} \right| = \frac{2}{3}\text{ and } \left( \vec{a} \times \vec{b} \right)\] is a unit vector. Write the angle between \[\vec{a} \text{ and } \vec{b}\] .
Solution
\[\text{ Given } : \vec{a} \times \vec{b} \text{ is a unit vector } .\]
\[ \Rightarrow \left| \vec{a} \times \vec{b} \right| = 1 . . . (1)\]
\[\text{ Let} \theta \text{ be the angle between } \vec{a} \text{ and } \vec{b} . \]
\[\text{ We know } \]
\[\left| \vec{a} \times \vec{b} \right| = \left| \vec{a} \right| \left| \vec{b} \right| \sin \theta\]
\[\text{ From (1), we get} \]
\[1 = \left( \sqrt{3} \right) \left( \frac{2}{3} \right) \sin \theta \]
\[ \Rightarrow \sin \theta = \frac{\sqrt{3}}{2}\]
\[ \Rightarrow \theta = \frac{\pi}{3}\]
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