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Question
If \[\vec{b}\] is a unit vector such that\[\left( \vec{a} + \vec{b} \right) . \left( \vec{a} - \vec{b} \right) = 8, \text{ find } \left| \vec{a} \right| .\]
Solution
\[\text{ Given that } \vec{b} \text{ is a unit vector }.\]
\[ \therefore \left| \vec{b} \right| = 1\]
\[\text{ And }\]
\[\left( \vec{a} + \vec{b} \right) . \left( \vec{a} - \vec{b} \right) = 8 ..........\left( \text{ Given } \right)\]
\[ \Rightarrow \left| \vec{a} \right|^2 - \left| \vec{b} \right|^2 = 8\]
\[ \Rightarrow \left| \vec{a} \right|^2 - 1^2 = 8\]
\[ \Rightarrow \left| \vec{a} \right|^2 = 9\]
\[ \therefore \left| \vec{a} \right| = 3\]
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