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

• A. \(\mathbf{i}\)
• B. \(\mathbf{k}\)
• C. \(\mathbf{j}\)
• D. \(\mathbf{i} + \mathbf{j} + \mathbf{k}\)

Hint:

Set the components equal and solve.

Answer 1

The Correct Option is A

To solve this problem, we must analyze the conditions given and understand what they imply about the vector \(\mathbf{a}\).

We are given the conditions:

• \(\mathbf{a} \cdot \mathbf{i} = \mathbf{a} \cdot (\mathbf{j} + \mathbf{i}) = \mathbf{a} \cdot (\mathbf{i} + \mathbf{j} + \mathbf{k})\)

Start with the expression \(\mathbf{a} \cdot \mathbf{i}\):

• Let \(\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}\)
• Then, \(\mathbf{a} \cdot \mathbf{i} = a_1\)

For the expression \(\mathbf{a} \cdot (\mathbf{j} + \mathbf{i})\):

• \(\mathbf{a} \cdot (\mathbf{j} + \mathbf{i}) = \mathbf{a} \cdot \mathbf{j} + \mathbf{a} \cdot \mathbf{i}\)
• This expands to \(a_2 + a_1\).
• From the condition, \(a_1 = a_2 + a_1\).
• This implies \(a_2 = 0\).

For the expression \(\mathbf{a} \cdot (\mathbf{i} + \mathbf{j} + \mathbf{k})\):

• \(\mathbf{a} \cdot (\mathbf{i} + \mathbf{j} + \mathbf{k}) = \mathbf{a} \cdot \mathbf{i} + \mathbf{a} \cdot \mathbf{j} + \mathbf{a} \cdot \mathbf{k}\)
• This expands to \(a_1 + a_2 + a_3 = a_1 + 0 + a_3 = a_1\).
• From the condition, \(a_1 = a_1 + a_3\).
• This implies \(a_3 = 0\).

Hence, based on the above deductions:

• \(\mathbf{a} = a_1\mathbf{i}\) (since both \(a_2 = 0\) and \(a_3 = 0\))
• Thus, \(\mathbf{a}\) is in the direction of \(\mathbf{i}\).

Therefore, the correct answer is \(\mathbf{i}\).