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Question
If \[\vec{a,} \vec{b}\] \[\text { are non-collinear vectors, then find the value of} \left[ \vec{a} \vec{b}\hat { i} \right] \hat{i} + \left[ \vec{a} \vec{b} \hat {j} \right] \hat {j} + \left[ \vec{a} \vec{b} \hat {k} \right] \hat {k} .\]
Solution
\[\text {For any vector }\vec{r} , \text {we have }\]
\[\left( \vec{r} \cdot \hat {i} \right) \hat {i} + \left( \vec{r} \cdot \hat {j} \right) \hat {j} + \left( \vec{r} \cdot \hat {k} \right) \hat {k} = \vec{r} \]
\[\text { Replacing } \vec{r} \text { by } \vec{a} \times \vec{b} , \text { we have }\]
\[ \left[ \left( \vec{a} \times \vec{b} \right) \cdot\hat { i} \right] \hat {i}+ \left[ \left( \vec{a} \times \vec{b} \right) \cdot \hat {j} \right] \hat {j} + \left[ \left( \vec{a} \times \vec{b} \right) \cdot \hat {k} \right] \hat{k} = \vec{a} \times \vec{b} \]
\[ \therefore \left[ \vec{a} \vec{b} \hat {i} \right] \hat {i} + \left[ \vec{a} \vec{b}\hat { j } \right]\hat { j} + \left[ \vec{a} \vec{b} \hat {k } \right] \hat {k} = \vec{a} \times \vec{b} \]
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