If \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{0}},|\overrightarrow{\mathbf{a}}|=3,|\overrightarrow{\mathbf{b}}|=5,|\overrightarrow{\mathbf{c}}|=7\), then the angle between \(\overrightarrow{\mathbf{a}}\) and \(\overrightarrow{\mathbf{b}}\) is

Given, \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{0}}\)
\(\therefore \overrightarrow{\mathbf{c}}=-(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}) \)
\( \Rightarrow |\overrightarrow{\mathbf{c}}|^{2}=\overrightarrow{\mathbf{c}} \overrightarrow{\mathbf{c}}=(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}) \cdot(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}})\)
\(\Rightarrow |\overrightarrow{\mathbf{c}}|^{2}=|\overrightarrow{\mathbf{a}}|^{2}+|\overrightarrow{\mathbf{b}}|^{2}+2 \overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}} \)
\( \Rightarrow |\overrightarrow{\mathbf{c}}|^{2}=|\overrightarrow{\mathbf{a}}|^{2}+|\overrightarrow{\mathbf{b}}|^{2}+2|\overrightarrow{\mathbf{a}}||\overrightarrow{\mathbf{b}}| \cos \theta \)
\( \Rightarrow 49=9+25+2 \times 3 \times 5 \cos \theta \)
\( \Rightarrow 15=30 \cos \theta \Rightarrow \cos \theta=\frac{1}{2} \)
\( \Rightarrow \theta=\frac{\pi}{3}\)