Question

If $\hat{a},\hat{b}$ are unit vectors such that $\hat{a}+\hat{b}$is a unit vector, write the value of $\left|\hat{a}-\hat{b}\right|.$

Answer 1

$$\begin{gathered} \text{Given that}\hat{\text{a}}\text{and}\hat{b}\text{are unit vectors such that}\hat{a}\text{+}\hat{b}\text{is a unit vector.} \\ \Rightarrow \left|\hat{\text{a}}\right|=\left|\hat{b}\right|=\left|\hat{\text{a}}+\hat{b}\right|=1...\left(1\right) \\ \text{Now,} \\ \left|\hat{\text{a}}+\hat{b}\right|=1 \\ \text{Squaring both sides, we get} \\ {\left|\hat{\text{a}}\right|}^2+{\left|\hat{b}\right|}^2+2\hat{\text{a}}.\hat{b}=1 \\ \Rightarrow 1+1+2\hat{\text{a}}.\hat{b}=1\left[\text{From (1)}\right] \\ \Rightarrow \hat{\text{a}}.\hat{b}=\frac{-1}{2}...\left(2\right) \\ \text{Now,} \\ {\left|\hat{\text{a}}-\hat{b}\right|}^2={\left|\hat{\text{a}}\right|}^2+{\left|\hat{b}\right|}^2-2\hat{\text{a}}.\hat{b} \\ =1+1-2\left(\frac{-1}{2}\right)=3\left[\text{From}(1)\mathrm{and}\left(2\right)\right] \\ \therefore \left|\hat{\text{a}}-\hat{b}\right|=\sqrt{3} \end{gathered}$$