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Question
Find \[\left| \vec{a} \right| \text{ and } \left| \vec{b} \right|\] if
\[\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) = 8 \text{ and } \left| \vec{a} \right| = 8\left| \vec{b} \right|\]
Solution
\[\ \text{ Given that }\]
\[\text{ And } \left| \vec{a} \right| = 8 \left| \vec{b} \right| . . . \left( 1 \right)\]
\[\left( \vec{a} + \vec{b} \right) . \left( \vec{a} - \vec{b} \right) = 8\]
\[ \Rightarrow \left| \vec{a} \right|^2 - \left| \vec{b} \right|^2 = 8\]
\[ \Rightarrow \left( 8 \left| \vec{b} \right| \right)^2 - \left| \vec{b} \right|^2 = 8............. \left[ \text{ From } (1) \right]\]
\[ \Rightarrow 64 \left| \vec{b} \right|^2 - \left| \vec{b} \right|^2 = 8\]
\[ \Rightarrow 63 \left| \vec{b} \right|^2 = 8\]
\[ \Rightarrow \left| \vec{b} \right|^2 = \frac{8}{63}\]
\[ \Rightarrow \left| \vec{b} \right| = \sqrt{\frac{8}{63}}\]
\[\left| \vec{a} \right| = 8 \left| \vec{b} \right| = 8 \sqrt{\frac{8}{63}} = \frac{8\sqrt{8}}{\sqrt{63}}\]
\[ \therefore \left| \vec{a} \right| = \frac{8\sqrt{8}}{\sqrt{63}} \text{ and } \left| \vec{b} \right| = \sqrt{\frac{8}{63}}\]
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