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Question
Write the value of \[\hat{ i } . \left( \hat{ j } \times \hat{ k } \right) + \hat{ j } . \left( \hat{ k } \times \hat{ i } \right) + \hat{ k } . \left( \hat{ j } \times \hat{ i } \right) .\]
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Solution
\[\hat{ i } . \left( \hat{ j } \times \hat{ k } \right) + \hat{ j } . \left( \hat{ k } \times \hat{ i } \right) + \hat{ k } . \left( \hat{ j } \times \hat{ i } \right)\]
\[ = \hat{ i } . \hat{ i} + \hat{ j} . j + \hat{ k } . \left( - \hat{ k } \right)\]
\[ = \left| \hat{ i } \right|^2 + \left| \hat{ j } \right|^2 - \left| \hat{ k } \right|^2 \]
\[ = 1 + 1 - 1 ( \because \left| \hat{ i } \right| = 1, \left| \hat{ j } \right| = 1 \text{ and } \left| \hat{ k } \right| = 1)\]
\[ = 1\]
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