Question:
If \(\mathbf{a} \cdot \mathbf{i} = \mathbf{a} \cdot (\mathbf{j} + \mathbf{i}) = \mathbf{a} \cdot (\mathbf{i} + \mathbf{j} + \mathbf{k})\), then \(\mathbf{a}\) is equal to
If \(\mathbf{a} \cdot \mathbf{i} = \mathbf{a} \cdot (\mathbf{j} + \mathbf{i}) = \mathbf{a} \cdot (\mathbf{i} + \mathbf{j} + \mathbf{k})\), then \(\mathbf{a}\) is equal to
Show Hint
Set the components equal and solve.
Updated On: Apr 23, 2026
- \(\mathbf{i}\)
- \(\mathbf{k}\)
- \(\mathbf{j}\)
- \(\mathbf{i} + \mathbf{j} + \mathbf{k}\)
Show Solution
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The Correct Option is A
Solution and Explanation
Step 1: Formula / Definition}
\[ \mathbf{a} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \]
Step 2: Calculation / Simplification}
\(\mathbf{a} \cdot \mathbf{i} = x\)
\(\mathbf{a} \cdot (\mathbf{i} + \mathbf{j}) = x + y\)
\(\mathbf{a} \cdot (\mathbf{i} + \mathbf{j} + \mathbf{k}) = x + y + z\)
\(x = x + y = x + y + z\)
\(x = x + y \Rightarrow y = 0\)
\(x + y = x + y + z \Rightarrow z = 0\)
Let \(x = 1\) (unit vector): \(\mathbf{a} = \mathbf{i}\)
Step 3: Final Answer
\[ \mathbf{i} \]
\[ \mathbf{a} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \]
Step 2: Calculation / Simplification}
\(\mathbf{a} \cdot \mathbf{i} = x\)
\(\mathbf{a} \cdot (\mathbf{i} + \mathbf{j}) = x + y\)
\(\mathbf{a} \cdot (\mathbf{i} + \mathbf{j} + \mathbf{k}) = x + y + z\)
\(x = x + y = x + y + z\)
\(x = x + y \Rightarrow y = 0\)
\(x + y = x + y + z \Rightarrow z = 0\)
Let \(x = 1\) (unit vector): \(\mathbf{a} = \mathbf{i}\)
Step 3: Final Answer
\[ \mathbf{i} \]
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