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Question
If \[\vec{a} \text{ and } \vec{b}\] are vectors of equal magnitude, write the value of \[\left( \vec{a} + \vec{b} \right) . \left( \vec{a} - \vec{b} \right) .\]
Solution
\[\text{ We have }\]
\[\left| \vec{a} \right| = \left| \vec{b} \right| . . . \left( i \right)\]
\[\text{ Now }, \]
\[\left( \vec{a} + \vec{b} \right) . \left( \vec{a} - \vec{b} \right)\]
\[ = \left| \vec{a} \right|^2 - \left| \vec{b} \right|^2 \]
\[ = \left| \vec{a} \right|^2 - \left| \vec{a} \right|^2 ...................\left[ \text{ Using }\left( i \right) \right]\]
\[ = 0\]
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