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Question
Write the value of \[\left[ \hat {i} - \hat {j} \hat {j} - \hat {k} \hat {k} - \hat {i} \right] .\]
Solution
We have
\[\left[\hat { i} - \hat {j} \hat {j} - \hat {k} \hat {k} - \hat {i} \right] = \left[ \left( \hat {i} - \hat {j} \right) \times \left( \hat {j} - \hat {k} \right) \right] \cdot \left( \hat{k} - \hat {i} \right) \left( \because \left[ \vec{a} \vec{b} \vec{c} \right] = \left( \vec{a} \times \vec{b} \right) . \vec{c} \right)\]
\[ = \left[ \left( \hat {i} \times \hat {j} \right) - \left( \hat {i} \times \hat {k} \right) - \left(\hat { j} \times \hat {j} \right) + \left( \hat {j} \times \hat {k} \right) \right] \cdot \left(\hat { k} - ]\hat {i} \right)\]
\[ = \left[ \hat {k} + \hat {j} + \hat {i} \right] \cdot \left(\hat { k} - \hat {i} \right)\]
\[ = \left[ \left( \hat {k} \cdot \hat {k} \right) - \left( \hat {k} \cdot\hat { i } \right) + \left( \hat {j} \cdot \hat {k} \right) - \left( \hat {j} \cdot \hat {i} \right) + \left( \hat {i} \cdot \hat {k} \right) - \left( \hat {i} \cdot \hat {i} \right) \right]\]
\[ = 1 - 0 + 0 - 0 + 0 - 1 = 0\]
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