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Question
If \[\vec{a} \text{ and } \vec{b}\] are two vectors such that \[\left| \vec{a} \times \vec{b} \right| = \sqrt{3}\text{ and } \vec{a} . \vec{b} = 1,\] find the angle between.
Solution
\[\left| \vec{a} \times \vec{b} \right| = \sqrt{3}\]
\[ \Rightarrow \left| \vec{a} \right| \left| \vec{b} \right| \sin \theta = \sqrt{3} . . . (1)\]
\[ \vec{a} . \vec{b} = 1\]
\[ \Rightarrow \left| \vec{a} \right| \left| \vec{b} \right| \cos \theta = 1 . . . (2)\]
\[\text{ Dividing (1) by (2), we get } \]
\[\frac{\left| \vec{a} \right| \left| \vec{b} \right| \sin \theta}{\left| \vec{a} \right| \left| \vec{b} \right| \cos \theta} = \sqrt{3}\]
\[ \Rightarrow \tan \theta = \sqrt{3}\]
\[ \Rightarrow \theta = {60}^o \]
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