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Question
If \[\left| \vec{a} \times \vec{b} \right|^2 + \left( \vec{a} . \vec{b} \right)^2 = 144\] and \[\left| \vec{a} \right| = 4,\] find \[\left| \vec{b} \right|\] .
Solution
\[\text{ We know} \]
\[ \left| \vec{a} \times \vec{b} \right|^2 + \left( \vec{a} . \vec{b} \right)^2 = \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 \]
\[ \Rightarrow 144 = 4^2 \left| \vec{b} \right|^2 \]
\[ \Rightarrow 144 = 16 \left| \vec{b} \right|^2 \]
\[ \Rightarrow \left| \vec{b} \right|^2 = 9\]
\[ \Rightarrow \left| \vec{b} \right| = 3\]
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