Let \(\mathbf{a}\) and \(\mathbf{b}\) be two unit vectors and \(\theta\) is the angle between them. Then, \(\mathbf{a}+\mathbf{b}\) is a unit vector, if

\(\because \mathbf{a}+\mathbf{b}\) is a unit vector if \(|\mathbf{a}+\mathbf{b}|=1\)
\(\Rightarrow \quad|\mathbf{a}+\mathbf{b}|^2=1\)
\(\Rightarrow \quad(\mathbf{a}+\mathbf{b}) \cdot(\mathbf{a}+\mathbf{b})=1\)
\(\Rightarrow \quad|\mathbf{a}|^2+|\mathbf{b}|^2+2(\mathbf{a} \cdot \mathbf{b})=1\)
\(\Rightarrow \quad 1+1+2(a \cdot b)=1\)
\(\therefore \quad 2(\mathbf{a} \cdot \mathbf{b})=-1\)
\(\Rightarrow \quad \mathbf{a} \cdot \mathbf{b}=-\frac{1}{2}\)
\(\Rightarrow \quad|\mathbf{a}||\mathbf{b}| \cos \theta=-\frac{1}{2}\)
\(\Rightarrow \quad 1 \times 1 \times \cos \theta=-\frac{1}{2}\)
\(\Rightarrow \quad \cos \theta=-\frac{1}{2}\)
Hence, \(\theta=\frac{2 \pi}{3}\)