A unit vector perpendicular to the vectors $î-ĵ$ and $î +ĵ$ is

$\hat k$

The correct answer is Option (1) → $\hat k$

Given vectors: $\vec{a} = \hat{i} - \hat{j}$ and $\vec{b} = \hat{i} + \hat{j}$

To find a unit vector perpendicular to both $\vec{a}$ and $\vec{b}$, take the cross product:

$\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -1 & 0 \\ 1 & 1 & 0 \end{vmatrix}$

$= \hat{i}( -1 \cdot 0 - 0 \cdot 1 ) - \hat{j}(1 \cdot 0 - 0 \cdot 1) + \hat{k}(1 \cdot 1 - (-1) \cdot 1)$

$= \hat{i}(0) - \hat{j}(0) + \hat{k}(1 + 1) = 2\hat{k}$

Unit vector = $\frac{2\hat{k}}{|2\hat{k}|} = \frac{2\hat{k}}{2} = \hat{k}$