Question

If \[\vec{a} \text{ and } \vec{b}\] are two vectors such that \[\left( \vec{a} + \vec{b} \right) . \left( \vec{a} - \vec{b} \right) = 0,\] find the relation between the magnitudes of \[\vec{a} \text{ and } \vec{b}\]  

Answer 1

\[\text{ Given that }\]
\[\left( \vec{a} + \vec{b} \right) . \left( \vec{a} - \vec{b} \right) = 0\]
\[ \Rightarrow \left| \vec{a} \right|^2 - \left| \vec{b} \right|^2 = 0\]
\[ \Rightarrow \left| \vec{a} \right|^2 = \left| \vec{b} \right|^2 \]
\[ \therefore \left| \vec{a} \right| = \left| \vec{b} \right|\]