Question

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

\[\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) = 8 \text{ and } \left| \vec{a} \right| = 8\left| \vec{b} \right|\]

Answer 1

\[\ \text{ Given that }\]

\[\text{ And } \left| \vec{a} \right| = 8 \left| \vec{b} \right| . . . \left( 1 \right)\]

\[\left( \vec{a} + \vec{b} \right) . \left( \vec{a} - \vec{b} \right) = 8\]

\[ \Rightarrow \left| \vec{a} \right|^2 - \left| \vec{b} \right|^2 = 8\]

\[ \Rightarrow \left( 8 \left| \vec{b} \right| \right)^2 - \left| \vec{b} \right|^2 = 8............. \left[ \text{ From } (1) \right]\]

\[ \Rightarrow 64 \left| \vec{b} \right|^2 - \left| \vec{b} \right|^2 = 8\]

\[ \Rightarrow 63 \left| \vec{b} \right|^2 = 8\]

\[ \Rightarrow \left| \vec{b} \right|^2 = \frac{8}{63}\]

\[ \Rightarrow \left| \vec{b} \right| = \sqrt{\frac{8}{63}}\]

\[\left| \vec{a} \right| = 8 \left| \vec{b} \right| = 8 \sqrt{\frac{8}{63}} = \frac{8\sqrt{8}}{\sqrt{63}}\]

\[ \therefore \left| \vec{a} \right| = \frac{8\sqrt{8}}{\sqrt{63}} \text{ and } \left| \vec{b} \right| = \sqrt{\frac{8}{63}}\]