If a · i = a · (i + j) = a · (i + j + k) = 1, then a is

The correct answer is Option (1) → $\hat{i}$ ##

Given that $\vec{a} \cdot \hat{i} = \vec{a} \cdot (\hat{i} + \hat{j}) = \vec{a} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1$

Assume that $\vec{a} = x\hat{i} + y\hat{j} + z\hat{k}$

So that, $\vec{a} \cdot \hat{i} = x = 1$

$\vec{a} \cdot (\hat{i} + \hat{j}) = x + y = 1$

$\vec{a} \cdot (\hat{i} + \hat{j} + \hat{k}) = x + y + z = 1$

This implies $x = 1, x + y = 1, x + y + z = 1$

By solving the equation, $x = 1, y = 0, z = 0$

Thus, $\vec{a} = \hat{i}$