If the sum of two unit vectors is unit vector then the magnitude of their difference is

$$\begin{gathered} if \widehat{A}.\widehat{B} are unit vectors then \widehat{A}+\widehat{B}=\widehat{C} \\ \therefore {\text{A}}^{\text{2}}+{\text{B}}^2+\text{AB}\cos \theta ={\text{C}}^{\text{2}} \\ \text{But }\left|\widehat{A}\right|=\left|\widehat{B}\right|=\left|\widehat{C}\right|=1 \\ \therefore 1+1+2\cos \theta =1 \\ \cos \theta =-\frac{1}{2} \\ Difference of unit vector \\ \left|\widehat{A}-\widehat{B}\right|=\sqrt{{\text{A}}^2+{\text{B}}^{\text{2}}-2\text{AB}\cos \theta } \\ =\sqrt{1+1-2\times 1\times 1\left(-\frac{1}{2}\right)} \\ =\sqrt{3} \end{gathered}$$