If $\mathbf{a} \cdot \hat{\mathbf{i}}=\mathbf{a} \cdot(\hat{\mathbf{i}}+\hat{\mathbf{j}})=\mathbf{a} \cdot(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})$, then $\mathbf{a}$ is equal to

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Aicharya · Accepted Answer

We have $ \begin{aligned} & \mathbf{a} \cdot \hat{\mathbf{i}}=\mathbf{a} \cdot(\hat{\mathbf{i}}+\hat{\mathbf{j}})=\mathbf{a} \cdot \hat{\mathbf{i}}+\mathbf{a} \cdot \hat{\mathbf{j}} 	ext { a } \hat{\mathbf{j}}=0 \ & 	ext { Also, } \quad \mathbf{a} \cdot \hat{\mathbf{i}}=\mathbf{a} \cdot(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}) \ & a \cdot \hat{\mathbf{i}}+\mathbf{a} \cdot \hat{\mathbf{j}}=\mathbf{a} \cdot \hat{\mathbf{i}}+\mathbf{a} \hat{\mathbf{j}}+\mathbf{a} \cdot \hat{\mathbf{k}} \ & \Rightarrow \quad \hat{\mathbf{a}} \cdot \hat{\mathbf{k}}=0 \ & 	ext { Also, } \quad \hat{\mathbf{i}} \cdot \hat{\mathbf{j}}=\hat{\mathbf{j}} \cdot \hat{\mathbf{k}}=\hat{\mathbf{k}} \cdot \hat{\mathbf{i}}=0 \ & 	ext { Then } \mathbf{a}=\hat{\mathbf{i}} \ & \end{aligned} $

If $\mathbf{a} \cdot \hat{\mathbf{i}}=\mathbf{a} \cdot(\hat{\mathbf{i}}+\hat{\mathbf{j}})=\mathbf{a}…

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