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Question
For any two vectors \[\vec{a} \text{ and } \vec{b}\] show that \[\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) = 0 \Leftrightarrow \left| \vec{a} \right| = \left| \vec{b} \right|\]
Solution
\[\text{ We have }\]
\[\left( \vec{a} + \vec{b} \right) . \left( \vec{a} - \vec{b} \right) = 0\]
\[ \Rightarrow \left| \vec{a} \right|^2 - \left| \vec{b} \right|^2 = 0\]
\[ \Rightarrow \left| \vec{a} \right|^2 = \left| \vec{b} \right|^2 \]
\[ \Rightarrow \left| \vec{a} \right| = \left| \vec{b} \right|\]
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