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Question
Write a vector satisfying \[\vec{a} . \hat{i} = \vec{a} . \left( \hat{i} + \hat{j} \right) = \vec{a} . \left( \hat{i} + \hat{j} + \hat{k} \right) = 1 .\]
Solution
\[\text{ Let } \vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \]
\[ \vec{a} . \hat{i} = a_1 \]
\[ \vec{a} . \left( \hat{i} + \hat{j} \right) = a_1 + a_2 \]
\[ \vec{a} . \left( \hat{i} + \hat{j} + \hat{k} \right) = a_1 + a_2 + a_3 \]
\[\text{ Given that }\]
\[ \vec{a} . \hat{i} = \vec{a} . \left( \hat{i} + \hat{j} \right) = \vec{a} . \left( \hat{i} + \hat{j} + \hat{k} \right) = 1\]
\[ \Rightarrow a_1 = a_1 + a_2 = a_1 + a_2 + a_3 = 1\]
\[ \Rightarrow a_1 = 1; a_1 + a_2 = 1; a_1 + a_2 + a_3 = 1\]
\[ \Rightarrow a_1 = 1; 1 + a_2 = 1; 1 + a_2 + a_3 = 1\]
\[ \Rightarrow a_1 = 1; a_2 = 0; 1 + 0 + a_3 = 1\]
\[ \Rightarrow a_1 = 1; a_2 = 0; a_3 = 0\]
\[So, \vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} = 1 \hat{i} + 0 \hat{j} + 0 \hat{k} = \hat{i} \]
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