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Question
If \[\hat{a} , \hat{b}\] are unit vectors such that \[\hat{a} + \hat{b}\] is a unit vector, write the value of \[\left| \hat{a} - \hat{b} \right| .\]
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Solution
\[\text{ Given that } \hat{a} \text{ and } \hat{b} \text{ are unit vectors such that } \hat{a} + \hat{b} \text{ is a unit vector }.\]
\[ \Rightarrow \left| \hat{a} \right| = \left| \hat{b} \right| = \left| \hat{a} + \hat{b} \right| = 1 . . . \left( 1 \right)\]
Now ,
\[\left| \hat{a} + \hat{b} \right| = 1\]
\[\text{ Squaring both sides, we get }\]
\[ \left| \hat{a} \right|^2 + \left| \hat{b} \right|^2 + 2 \hat{a} . \hat{b} = 1\]
\[ \Rightarrow 1 + 1 + 2 \hat{a} . \hat{b} = 1............. \left[ \text{ From (1) }\right]\]
\[ \Rightarrow \hat{a} . \hat{b} = \frac{- 1}{2} . . . \left( 2 \right)\]
\[\text{ Now },\]
\[ \left| \hat{a} - \hat{b} \right|^2 = \left| \hat{a} \right|^2 + \left| \hat{b} \right|^2 - 2 \hat{a} . \hat{b} \]
\[ = 1 + 1 - 2\left( \frac{- 1}{2} \right) = 3................. \left[ \text{ From } (1) \text{ and } (2) \right]\]
\[ \therefore \left| \hat{a} - \hat{b} \right| = \sqrt{3}\]
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