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Question
if \[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 7 \text{ and } \vec{a} \times \vec{b} = 3 \hat{ i } + 2 \hat{ j } + 6 \hat{ k } ,\] find the angle between \[\vec{a} \text{ and } \vec{b} .\]
Solution
\[\text{ Let } \theta \text{ be the angle between } \vec{a} \text{ and } \vec{b} .\]
\[ \vec{a} \times \vec{b} = 3 \hat{ i } + 2 \hat{ j } + 6 \hat{ k } (\text{ Given } )\]
\[ \Rightarrow \left| \vec{a} \times \vec{b} \right| = \sqrt{9 + 4 + 36}\]
\[ = 7\]
\[\text{ We know } \]
\[\left| \vec{a} \times \vec{b} \right| = \left| \vec{a} \right| \left| \vec{b} \right| \sin \theta\]
\[ \Rightarrow 7 = \left( 2 \right) \left( 7 \right) \sin \theta\]
\[ \Rightarrow \sin \theta = \frac{1}{2}\]
\[ \Rightarrow \theta = \frac{\pi}{6}\]
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