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Question
If \[\left| \vec{a} \times \vec{b} \right| = 4, \left| \vec{a} \cdot \vec{b} \right| = 2, \text{ then } \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 =\]
Options
6
2
20
8
Solution
\[\text{ We know } \]
\[ \left( \vec{a} . \vec{b} \right)^2 + \left| \vec{a} \times \vec{b} \right|^2 = \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 . . . (1)\]
\[\left| \vec{a} . \vec{b} \right| = 2 (\text{ Given } )\]
\[ \Rightarrow \left| \vec{a} . \vec{b} \right|^2 = \left( \vec{a} . \vec{b} \right)^2 \]
\[\text{ From (1), we get } \]
\[ \left( 2 \right)^2 + \left( 4 \right)^2 = \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 \]
\[ \Rightarrow \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 = 20\]
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