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Question
If \[\vec{a} \text{ and } \vec{b}\] are two vectors such that \[\left| \vec{a} . \vec{b} \right| = \left| \vec{a} \times \vec{b} \right|,\] write the angle between \[\vec{a} \text{ and } \vec{b} .\]
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Solution
\[\text{ Let } \theta \text{ be the angle between
} a^\to \text{ and } \vec{b} . \]
\[\text{ We know } \]
\[\left| \vec{a} \times \vec{b} \right| = \left| \vec{a} \right| \left| \vec{b} \right| \left| \sin \theta \right|\]
\[\left| \vec{a} . \vec{b} \right| = \left| \vec{a} \right| \left| \vec{b} \right|\left| \cos \theta \right|\]
\[\text{ Now, } \]
\[\left| \vec{a} \times \vec{b} \right| = \left| \vec{a} . \vec{b} \right| (\text{Given } )\]
\[ \Rightarrow \left| \vec{a} \right| \left| \vec{b} \right| \left| \sin \theta \right| = \left| \vec{a} \right| \left| \vec{b} \right| \left| \cos \theta \right|\]
\[ \Rightarrow \left| \sin \theta \right| = \left| \cos \theta \right|\]
\[ \Rightarrow \theta = \frac{\pi}{4}\]
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