दिया है \(\left|\overrightarrow{ A }_{1}\right|=3,\left|\overrightarrow{ A }_{2}\right|=5\) तथा \(\left|\overrightarrow{ A }_{1}+\overrightarrow{ A }_{2}\right|=5\) तो \(\left(2 \overrightarrow{ A }_{1}+3 \overrightarrow{ A }_{2}\right) \cdot\left(3 \overrightarrow{ A }_{1}-2 \overrightarrow{ A }_{2}\right)\) का मान होगा ?

\(\begin{array}{l}
\left| {{{\vec A}_1}} \right| = 3,\left| {{{\vec A}_2}} \right| = 5,\,and\,\left| {{{\vec A}_1} + {{\vec A}_2}} \right| = 5.\\
\left| {{{\vec A}_1} + {{\vec A}_2}} \right| = {\left| {{{\vec A}_1}} \right|^2} + {\left| {{{\vec A}_2}} \right|^2} + 2\left| {{{\vec A}_1}} \right|\left| {{{\vec A}_2}} \right|\,\cos \,\theta \\
\cos \,\theta \, =  - \frac{3}{{10}}\\
\left( {2{{\vec A}_1} + 3{{\vec A}_2}} \right).\left( {3{{\vec A}_1} - 2{{\vec A}_2}} \right)\\
 = 6{\left| {{{\vec A}_1}} \right|^2} + 9{{\vec A}_1}.{{\vec A}_2} - 4{{\vec A}_1}.{{\vec A}_2} - 6{\left| {{{\vec A}_2}} \right|^2}\\
 =  - 118.5
\end{array}\)