Question

Write the value of  \[\hat{ i } . \left( \hat{ j } \times \hat{ k }  \right) + \hat{ j }  . \left( \hat{ k } \times \hat{ i }  \right) + \hat{ k }  . \left( \hat{ j }  \times \hat{ i }  \right) .\]

 

Answer 1

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

\[ = 1\]