Question

\[\text{ If }  \left| \vec{a} \right| = 2, \left| \vec{b} \right| = 5 \text{ and }  \left| \vec{a} \times \vec{b} \right| = 8, \text { find }  \vec{a} \cdot \vec{b} .\]

 

Answer 1

\[\text{ We know } \]
\[ \left( \vec{a} . \vec{b} \right)^2 + \left| \vec{a} \times \vec{b} \right|^2 = \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 \]
\[ \Rightarrow \left( \vec{a} . \vec{b} \right)^2 + 8^2 = 2^2 \times 5^2 ( \because \left| \vec{a} \times \vec{b} \right| = 8, \left| \vec{a} \right| = 2 \text{ and }  \left| \vec{b} \right| = 5)\]
\[ \Rightarrow \left( \vec{a} . \vec{b} \right)^2 + 64 = 100\]
\[ \Rightarrow \left( \vec{a} . \vec{b} \right)^2 = 36\]
\[ \Rightarrow \left( \vec{a} . \vec{b} \right) = 6\]