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Question
For any two non-zero vectors, write the value of \[\frac{\left| \vec{a} + \vec{b} \right|^2 + \left| \vec{a} - \vec{b} \right|^2}{\left| \vec{a} \right|^2 + \left| \vec{b} \right|^2} .\]
Solution
\[\text{ We have }\]
\[\frac{\left| \vec{a} + \vec{b} \right|^2 + \left| \vec{a} - \vec{b} \right|^2}{\left| \vec{a} \right|^2 + \left| \vec{b} \right|^2}\]
\[ = \frac{\left| \vec{a} \right|^2 + \left| \vec{b} \right|^2 + 2 \vec{a} . \vec{b} + \left| \vec{a} \right|^2 + \left| \vec{b} \right|^2 - 2 \vec{a} . \vec{b}}{\left| \vec{a} \right|^2 + \left| \vec{b} \right|^2}\]
\[ = \frac{2\left( \left| \vec{a} \right|^2 + \left| \vec{b} \right|^2 \right)}{\left| \vec{a} \right|^2 + \left| \vec{b} \right|^2}\]
\[ = 2\]
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