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Question
If \[\left| \vec{a} + \vec{b} \right| = 60, \left| \vec{a} - \vec{b} \right| = 40 \text{ and } \left| \vec{b} \right| = 46, \text{ find } \left| \vec{a} \right|\]
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Solution
\[\text{ We know that }\]
\[ \left| \vec{a} + \vec{b} \right|^2 + \left| \vec{a} - \vec{b} \right|^2 = 2\left( \left| \vec{a} \right|^2 + \left| \vec{b} \right|^2 \right)\]
\[ \Rightarrow {60}^2 + {40}^2 = 2\left( \left| \vec{a} \right|^2 + {46}^2 \right) ..................(\text{ Given })\]
\[ \Rightarrow 3600 + 1600 = 2 \left| \vec{a} \right|^2 + 4232\]
\[ \Rightarrow 968 = 2 \left| \vec{a} \right|^2 \]
\[ \Rightarrow \left| \vec{a} \right|^2 = 484\]
\[ \Rightarrow \left| \vec{a} \right| = 22\]
\[\]
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