#### Question

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

#### Solution

\[\text{ Given that } \hat{a}\ , \hat{b}\ \text{ and }\left| \hat{a} + \hat{b} \right|\text{ are unit vectors }.\]

\[So,\left| \hat{a} \right|=1,\left| \hat{b} \right|=1 and\left| \left| \hat{a} + \hat{b} \right| \right|=1\]

\[\text{ We have }\]

\[ \left| \hat{a} + \hat{b} \right|^2 + \left| \hat{a} - \hat{b} \right|^2 = 2\left( \left| \hat{a} \right|^2 + \left| \hat{b} \right|^2 \right)\]

\[ \Rightarrow 1 + \left| \hat{a} - \hat{b} \right|^2 = 2\left( 1 + 1 \right)\]

\[ \Rightarrow 1 + \left| \hat{a} - \hat{b} \right|^2 = 4\]

\[ \Rightarrow \left| \hat{a} - \hat{b} \right|^2 = 3\]

\[ \Rightarrow \left| \hat{a} - \hat{b} \right| = \sqrt{3}\]