Question

\[\text{ If }  \left| \vec{a} \right| = \sqrt{26}, \left| \vec{b} \right| = 7 \text{ and }  \left| \vec{a} \times \vec{b} \right| = 35, \text{ find }  \vec{a} . \vec{b} .\]

 

Answer 1

\[\text{ We know } \]
\[ \left( \vec{a} . \vec{b} \right)^2 + \left| \vec{a} \times \vec{b} \right|^2 = \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 \]
\[ \Rightarrow \left( \vec{a} . \vec{b} \right)^2 + {35}^2 = \left( \sqrt{26} \right)^2 \times 7^2 ( \because \left| \vec{a} \times \vec{b} \right| = 35, \left| \vec{a} \right| = \sqrt{26} \text{ and }  \left| \vec{b} \right| = 7)\]
\[ \Rightarrow \left( \vec{a} . \vec{b} \right)^2 + 1225 = 1274\]
\[ \Rightarrow \left( \vec{a} . \vec{b} \right)^2 = 49\]
\[ \Rightarrow \left( \vec{a} . \vec{b} \right) = 7\]