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Question
The value of \[\hat{ i } \cdot \left( \hat{ j } \times \hat{ k } \right) + \hat{ j } \cdot \left( \hat{ i } \times \hat{ k } \right) + \hat{ k } \cdot \left( \hat{ i } \times \hat{ j } \right),\] is
Options
0
-1
1
3
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Solution
\[\hat{ i } . \left( \hat{ j } \times \hat{ k } \right) + \hat{ j} . \left( \hat{ i } \times \hat{ k } \right) + \hat{ k } . \left( \hat{ i } \times \hat{ j } \right)\]
\[ = \hat{ i } . \hat{ i } + \hat{ j } . \left( - \hat{ j } \right) + \hat{ k } . \hat{ k } \]
\[ = \left| \hat{ i } \right|^2 - \left| \hat{ j } \right|^2 + \left| \hat{ k } \right|^2 \]
\[ = 1 - 1 + 1\]
\[ = 1\]
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