Question

If the sum of two unit vectors is a unit vector prove that the magnitude of their difference is $\sqrt{3}$.

Answer 1

$$\begin{gathered} \text{Given that}\overbrace{a}\text{,}\overbrace{b}\text{and}\left|\hat{a}+\hat{b}\right|\text{are unit vectors.} \\ \text{So,}\left|\hat{a}\right|\text{=1,}\left|\hat{b}\right|\text{=1 and}\left|\left|\hat{a}+\hat{b}\right|\right|\text{=1} \\ \text{We have} \\ {\left|\hat{a}+\hat{b}\right|}^2+{\left|\hat{a}-\hat{b}\right|}^2=2\left({\left|\hat{a}\right|}^2+{\left|\hat{b}\right|}^2\right) \\ \Rightarrow 1+{\left|\hat{a}-\hat{b}\right|}^2=2\left(1+1\right) \\ \Rightarrow 1+{\left|\hat{a}-\hat{b}\right|}^2=4 \\ \Rightarrow {\left|\hat{a}-\hat{b}\right|}^2=3 \\ \Rightarrow \left|\hat{a}-\hat{b}\right|=\sqrt{3} \end{gathered}$$