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Question
If \[\vec{a} \text{ and } \vec{b}\] are mutually perpendicular unit vectors, write the value of \[\left| \vec{a} + \vec{b} \right| .\]
Solution
\[\vec{a} \text{ and } \vec{b} \text{ are unit vectors and they are perpendicular }.\]
\[ \Rightarrow \left| \vec{a} \right| = \left| \vec{b} \right| = 1; \vec{a} . \vec{b} = 0 . . . \left( i \right)\]
\[\text{ Now },\]
\[ \left| \vec{a} + \vec{b} \right|^2 = \left| \vec{a} \right|^2 + \left| \vec{b} \right|^2 + 2 \vec{a} . \vec{b} \]
\[ = 1 + 1 + 2 \left( 0 \right).................. \left[ \text{ Using } \left( i \right) \right]\]
\[ = 2 \]
\[ \therefore \left| \vec{a} + \vec{b} \right| = \sqrt{2}\]
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