Question

If \[\left| \vec{a} \times \vec{b} \right|^2 + \left( \vec{a} . \vec{b} \right)^2 = 144\]  and \[\left| \vec{a} \right| = 4,\]  find \[\left| \vec{b} \right|\] . 

 

 

 

Answer 1

\[\text{ We know} \]
\[ \left| \vec{a} \times \vec{b} \right|^2 + \left( \vec{a} . \vec{b} \right)^2 = \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 \]
\[ \Rightarrow 144 = 4^2 \left| \vec{b} \right|^2 \]
\[ \Rightarrow 144 = 16 \left| \vec{b} \right|^2 \]
\[ \Rightarrow \left| \vec{b} \right|^2 = 9\]
\[ \Rightarrow \left| \vec{b} \right| = 3\]