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Question
If \[\vec{a} \text{ and } \vec{b}\] are unit vectors, find the angle between \[\vec{a} + \vec{b} \text{ and } \vec{a} - \vec{b} .\]
Solution
\[\text{ We have }\]
\[\left| \vec{a} \right| = 1 \text{ and } \left| \vec{b} \right| = 1...............\left( i \right)\]
\[\text{ Now }, \]
\[\left( \vec{a} + \vec{b} \right) . \left( \vec{a} - \vec{b} \right) = \left| \vec{a} \right|^2 - \left| \vec{b} \right|^2 \]
\[ = 1^2 - 1^2................... \left[ \text{ Using } \left( i \right) \right]\]
\[ = 0\]
\[ \Rightarrow \vec{a} + \vec{b} \text{ and } \vec{a} - \vec{b} \text{ are perpendicular }.\]
\[ \text{ ∴ Angle between }\left( \vec{a} + \vec{b} \right) \text{ and } \left( \vec{a} - \vec{b} \right) \text{is 90 }^0 .\]
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