If a ∙ i = a ∙(j + i) = a ∙ (j + i + k), then a is equal to

Dear Student,

$$\begin{gathered} Given: \\ \overrightarrow{a}.\hat{i} = \overrightarrow{a}.\left(\hat{j} + \hat{i}\right) = \overrightarrow{a}.\left(\hat{j} + \hat{i} + \hat{k}\right) \\ \Rightarrow \cancel{\overrightarrow{a}.\hat{i}} = \overrightarrow{a}.\hat{j} + \cancel{\overrightarrow{a}.\hat{i}} = \overrightarrow{a}.\hat{j} + \cancel{\overrightarrow{a}.\hat{i}} + \overrightarrow{a}.\hat{k} \left[\because \overrightarrow{x}.\left(\overrightarrow{y} + \overrightarrow{z}\right) = \overrightarrow{x}.\overrightarrow{y} + \overrightarrow{x}.\overrightarrow{z}\right] \\ \Rightarrow 0 = \overrightarrow{a}.\hat{j} = \overrightarrow{a}.\hat{j} + \overrightarrow{a}.\hat{k} \\ \Rightarrow 0 = \overrightarrow{a}.\hat{j} = \overrightarrow{a}.\left(\hat{j} + \hat{k}\right) \\ \because \hat{j}, \hat{k} \ne 0, \\ \therefore \boxed{\overrightarrow{\mathbf{a}}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{0}} \end{gathered}$$​

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