Question

Find \[\left| \vec{a} \right| \text{ and } \left| \vec{b} \right|\] if 

\[\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) = 12 \text{ and } \left| \vec{a} \right| = 2\left| \vec{b} \right|\]

Answer 1

\[ \text{ Given that }\]

\[\left| \vec{a} \right| = 2 \left| \vec{b} \right| . . . \left( 1 \right)\]

\[\text{ And } \left( \vec{a} + \vec{b} \right) . \left( \vec{a} - \vec{b} \right) = 12\]

\[ \Rightarrow \left| \vec{a} \right|^2 - \left| \vec{b} \right|^2 = 12\]

\[ \Rightarrow \left( 2 \left| \vec{b} \right| \right)^2 - \left| \vec{b} \right|^2 = 12 ...........\left[ \text{ From } (1) \right]\]

\[ \Rightarrow 4 \left| \vec{b} \right|^2 - \left| \vec{b} \right|^2 = 12\]

\[ \Rightarrow 3 \left| \vec{b} \right|^2 = 12\]

\[ \Rightarrow \left| \vec{b} \right|^2 = 4\]

\[ \Rightarrow \left| \vec{b} \right| = 2\]

\[\left| \vec{a} \right| = 2 \left| \vec{b} \right| = 2\left( 2 \right) = 4\]

\[ \therefore \left| \vec{a} \right| = 4 \text{ and } \left| \vec{b} \right| = 2\]