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Question
Evaluate the following:
\[\left[\hat{i}\hat{j}\hat{k} \right] + \left[ \hat{j}\hat{k}\hat {i} \right] + \left[ \hat{k}\hat{i} \hat{j} \right]\]
Solution
i) We have
\[\left[ \hat{i}\hat{j}\hat{k} \right] + \left[ \hat{j}\hat {k} \hat{i} \right] + \left[ \hat{k} \hat{i} \hat{j} \right]\]
\[ = \left[ \hat{i}\hat{j}\hat{ k} \right] + \left[ \hat{i} \hat{j} \hat{k} \right] + \left[ \hat{i}\hat{ j}\hat{ k} \right] \left( \because \left[ \vec{a} \vec{b} \vec{c} \right] = \left[ \vec{b} \vec{c} \vec{a} \right] = \left[ \vec{c} \vec{a} \vec{b} \right] \right)\]
\[ = 3 \left[ \hat{i} \hat{j}\hat{ k} \right]\]
\[ = 3\left(\hat{i} \times \hat{j} \right) . \hat{k} \left( \because \left[ \vec{a} \vec{b} \vec{c} \right] = \left( \vec{a} \times \vec{b} \right) . \vec{c} \right)\]
\[ = 3\left( \hat{k} . \hat{k} \right)\]
\[ = 3\left( 1 \right) = 3\]
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