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Question
If \[\left| \vec{a} \times \vec{b} \right|^2 + \left| \vec{a} \cdot \vec{b} \right|^2 = 400\] and \[\left| \vec{a} \right| = 5,\] then write the value of \[\left| \vec{b} \right| .\]
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Solution
\[\left| \vec{a} \times \vec{b} \right|^2 + \left| \vec{a} \cdot \vec{b} \right|^2 = 400\]
\[ \Rightarrow \left\{ \left| \vec{a} \right|\left| \vec{b} \right|\sin\theta \right\}^2 + \left\{ \left| \vec{a} \right|\left| \vec{b} \right|\cos\theta \right\}^2 = 400\]
\[ \Rightarrow \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 \sin^2 \theta + \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 \cos^2 \theta = 400\]
\[ \Rightarrow \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 = 400\]
\[ \Rightarrow 25 \times \left| \vec{b} \right|^2 = 400\]
\[ \Rightarrow \left| \vec{b} \right|^2 = 16\]
\[ \Rightarrow \left| \vec{b} \right| = 4\]
\[\]
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