Question

If \[\vec{a} \cdot \text{i} = \vec{a} \cdot \left( \hat{i} + \hat{j} \right) = \vec{a} \cdot \left( \hat{i} + \hat{j} + \hat{k} \right) = 1,\]  then \[\vec{a} =\] 

Options

• \[\vec{0}\] 

•  \[\hat{i}\]  

•   \[\hat{j}\]

• \[\hat{i} + \hat{j} + \hat{k}\] 

Answer 1

\[\hat{i}\]  

\[\text{ Let } \vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \]
\[ \vec{a} . \hat{i} = a_1 \]
\[\text{ and } \vec{a} . \left( \hat{i} + \hat{j} \right) = a_1 + a_2 \]
\[\text{ and } \vec{a} . \left( \hat{i} + \hat{j} + \hat{k} \right) = a_1 + a_2 + a_3 \]
\[\text{ Given },\]
\[ \vec{a} . \hat{i} = \vec{a} . \left( \hat{i} + \hat{j} \right) = \vec{a} . \left( \hat{i} + \hat{j} + \hat{k} \right) = 1\]
\[ \Rightarrow a_1 = a_1 + a_2 = a_1 + a_2 + a_3 = 1\]
\[ \Rightarrow a_1 = 1, a_2 = 0, a_3 = 0\]
\[So, \vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} = 1 \hat{i} + 0 \hat{j} + 0 \hat{k} = \hat{i}\]