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If $\overrightarrow{\mathbf{a}} \cdot \hat{\mathbf{i}}=\overrightarrow{\mathbf{a}} \cdot(\hat{\mathbf{i}}+\hat{\mathbf{j}})=\overrightarrow{\mathbf{a}} \cdot(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})=1$ then $\overrightarrow{\mathbf{a}}$ is equal to
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$\hat{\mathbf{i}}$
Let $\overrightarrow{\mathbf{a}}=\mathrm{a}_{1} \hat{\mathbf{i}}+\mathrm{a}_{2} \hat{\mathbf{j}}+\mathrm{a}_{3} \hat{\mathbf{k}}$
Given, $\overrightarrow{\mathbf{a}} \cdot \hat{\mathbf{i}}=\overrightarrow{\mathbf{a}} \cdot(\hat{\mathbf{i}}+\hat{\mathbf{j}})=\overrightarrow{\mathbf{a}} \cdot(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})=1$
$\therefore \quad \mathrm{a}_{1}=\mathrm{a}_{1}+\mathrm{a}_{2}=\mathrm{a}_{1}+\mathrm{a}_{2}+\mathrm{a}_{3}=1$
$\Rightarrow \quad \mathrm{a}_{1}=1, \mathrm{a}_{2}=0, \mathrm{a}_{3}=0$
$\therefore \quad \overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}$
Given, $\overrightarrow{\mathbf{a}} \cdot \hat{\mathbf{i}}=\overrightarrow{\mathbf{a}} \cdot(\hat{\mathbf{i}}+\hat{\mathbf{j}})=\overrightarrow{\mathbf{a}} \cdot(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})=1$
$\therefore \quad \mathrm{a}_{1}=\mathrm{a}_{1}+\mathrm{a}_{2}=\mathrm{a}_{1}+\mathrm{a}_{2}+\mathrm{a}_{3}=1$
$\Rightarrow \quad \mathrm{a}_{1}=1, \mathrm{a}_{2}=0, \mathrm{a}_{3}=0$
$\therefore \quad \overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}$
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