\(\overline{\mathrm{a}}, \overline{\mathrm{b}}\) and \(\overline{\mathrm{c}}\) are three vectors such that \(\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}=\overline{0}\) and \(|\overline{\mathrm{a}}|=3,|\overline{\mathrm{b}}|=5,|\overline{\mathrm{c}}|=7\), then the angle between \(\overline{\mathrm{a}}\) and \(\overline{\mathrm{b}}\) is

\(\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}=0 \Rightarrow \overline{\mathrm{c}}=-(\overline{\mathrm{a}}+\overline{\mathrm{b}})\) and let angle between \(\overline{\mathrm{a}}\) and \(\overline{\mathrm{b}}\) be \(\theta\)
\(\therefore|\overline{\mathrm{c}}|^2=(\overline{\mathrm{a}}+\overline{\mathrm{b}})^2=|\overline{\mathrm{a}}|^2+|\overline{\mathrm{b}}|^2+2 \overline{\mathrm{a}} \cdot \overline{\mathrm{b}} \)
\( =|\overline{\mathrm{a}}|^2+|\overline{\mathrm{b}}|^2+2|\overline{\mathrm{a}}| \cdot|\overline{\mathrm{b}}| \cdot \cos \theta \)
\( \therefore(7)^2=(3)^2+(5)^2+2(3)(5) \cos \theta \)
\( \therefore 49=9+25+30 \cos \theta \Rightarrow \cos \theta=\frac{15}{30}=\frac{1}{2} \Rightarrow \theta=\frac{\pi}{3}\)