Question

If \[\vec{a}\] is a unit vector, then find \[\left| \vec{x} \right|\]  in each of the following. 

\[\left( \vec{x} - \vec{a} \right) \cdot \left( \vec{x} + \vec{a} \right) = 8\] 

Answer 1

\[\text{ Given that } \vec{a} \text{ is a unit vector }.\]

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

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

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

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

\[ \Rightarrow \left| \vec{x} \right|^2 = 9\]

\[ \Rightarrow \left| \vec{x} \right| = 3\]