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Question
Evaluate the following:
\[\left[ 2 \hat{i}\hat{ j}\ \hat{k}\right] + \left[\hat{i}\hat{ k}\hat {j} \right] + \left[\hat{ k}\hat{ j} 2\hat{ i} \right]\]
Solution
ii ) We have
\[\left[ 2\hat{ i} \hat{j}\hat{ k} \right] + \left[ \hat{i} \hat{k}\hat{ j} \right] + \left[ \hat{k}\hat{ j} 2 \hat{i} \right]\]
\[ = 2 \left[\hat{ i} \hat{j}\hat{ k} \right] + \left[\hat{ i} \hat{k}\hat{ j} \right] + 2\left[ \hat{k}\hat{j}\hat{ i} \right] \left( \because \left[ l \ \vec{a} \ m \ \vec{b} \ n \ \vec{c} \right] = lmn \left[ \vec{a} \vec{b} \vec{c} \right] \right)\]
\[ = 2\left(\hat{i} \times \hat{j} \right) . \hat{k} + \left( \hat{i} \times \hat{k} \right) . \hat{j}+ 2\left(\hat{k} \times \hat{j} \right) . \hat{i} \left( \because \left[ \vec{a} \vec{b} \vec{c} \right] = \left( \vec{a} \times \vec{b} \right) . \vec{c} \right)\]
\[ = 2\left( \hat{k} . \hat{k}\right) + \left( -\hat{ j} . \hat{j} \right) + 2\left( - \hat{i}. \hat{i} \right)\]
\[ = 2\left( 1 \right) + \left( - 1 \right) + 2\left( - 1 \right)\]
\[ = 2 - 1 - 2 = - 1\]
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