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

Given, \(\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0}\)
and \(|\mathbf{a}|=5,|\mathbf{b}|=3,|\mathbf{c}|=7\)
\(\Rightarrow \quad \mathbf{a}+\mathbf{b}=-\mathbf{c}\)
On squaring both sides, we get \((\mathbf{a}+\mathbf{b})^{2}=(-\mathbf{c})^{2}\)
\(\Rightarrow\)
\(|\mathbf{a}+\mathbf{b}|^{2}=|\mathbf{c}|^{2}\)
\(\Rightarrow|\mathbf{a}|^{2}+|\mathbf{b}|^{2}+2 \mathbf{a} \cdot \mathbf{b}=|\mathbf{c}|^{2}\)
\((\because \theta\) be the angle between \(\mathbf{a}\) and \(\mathbf{b}\) )
\(\Rightarrow(5)^{2}+(3)^{2}+2|\mathbf{a}||\mathbf{b}| \cos \theta=(7)^{2}\)
\(\Rightarrow \quad 25+9+2 \cdot 5 \cdot 3 \cdot \cos \theta=49\)
\(\Rightarrow \quad 30 \cos \theta=15\)
\(\Rightarrow \quad \cos \theta=\frac{1}{2}=\cos 60^{\circ}\)
\(\Rightarrow \quad \theta=\frac{\pi}{3}\)