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Question
Find the angle between two vectors \[\vec{a} \text{ and } \vec{b}\] , if \[\left| \vec{a} \times \vec{b} \right| = \vec{a} \cdot \vec{b} .\]
Solution
\[\text{ Let } \theta \text{ be the angle between } \vec{a} \text{ and } \vec{b} . \]
\[\text{ Given } :\]
\[\left| \vec{a} \times \vec{b} \right| = \vec{a} . \vec{b} \]
\[ \Rightarrow \left| \vec{a} \right| \left| \vec{b} \right| \sin \theta = \left| \vec{a} \right| \left| \vec{b} \right| \cos \theta\]
\[ \Rightarrow \sin \theta = \cos \theta\]
\[ \Rightarrow \tan \theta = 1\]
\[ \Rightarrow \theta = \frac{\pi}{4}\]
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