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Question
If \[\vec{a} \text{ and } \vec{b}\] are unit vectors such that \[\vec{a} \times \vec{b}\] is also a unit vector, find the angle between \[\vec{a} \text{ and } \vec{b}\] .
Solution
\[\text { Let } \theta \text{ be the angle between } \vec{a} \text{ and } \vec{b} . \]
\[\text{ Given } :\]
\[\left| \vec{a} \times \vec{b} \right| = 1\]
\[\left| \vec{a} \right| = 1\]
\[\left| \vec{b} \right| = 1\]
\[\text{ We know } \]
\[\left| \vec{a} \times \vec{b} \right| = \left| \vec{a} \right| \left| \vec{b} \right| \sin \theta\]
\[ \Rightarrow 1 = \left( 1 \right) \left( 1 \right) \sin \theta\]
\[ \Rightarrow \sin \theta = 1\]
\[ \Rightarrow \theta = \frac{\pi}{2}\]
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