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Question
Write the value of \[\hat{ i } × \left( \hat{ j } + \hat{ k } \right) + \hat{ j } × \left( \hat{ k } + \hat{ i } \right) + \hat{ k } × \left( \hat{ i } + \hat{ j } \right) .\]
Solution
\[\hat{ i } \times \left( \hat{ j } + \hat{ k } \right) + \hat{ j } \times \left( \hat{ k } + \hat{ i } \right) + \hat{ k } \times \left( \hat{ i } + \hat{ j } \right)\]
\[ = \left( \hat{ i } \times \hat{ j } \right) + \left( \hat{ i } \times \hat{ k } \right) + \left( \hat{ j } \times \hat{ k } \right) + \left( \hat{ j } \times \hat{ i} \right) + \left( \hat{ k } \times \hat{ i } \right) + \left( \hat{ k } \times \hat{ j } \right)\]
\[ = \hat{ k } - \hat{ j } + \hat{ i } - \hat{ k } + \hat{ j } - \hat{ i } \]
\[ = \vec{0}\]
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