Question

Write the value of \[\left( \vec{a} . \hat{i} \right) \hat{i} + \left( \vec{a} . \hat{j} \right) \hat{j} + \left( \vec{a} . \hat{k} \right) \hat{k} ,\]  where \[\vec{a}\] is any vector. 

Answer 1

\[\text{ Let } \vec{a} = a_1 \vec{i} + a_2 \vec{j} + a_3 \vec{k} \]
\[\text{ Now },\]
\[\left( \vec{a} . \vec{i} \right) \vec{i} + \left( \vec{a} . \vec{j} \right) \vec{j} + \left( \vec{a} . \vec{k} \right) \vec{k} \]
\[ = a_1 \vec{i} + a_2 \vec{j} + a_3 \vec{k} \]
\[ = \vec{a}\]