The area of the rectangle having vertices $\mathrm{P}, \mathrm{Q}, \mathrm{R}, \mathrm{S}$ with position vectors $-\hat{i}+\hat{j}+\hat{k}, \hat{i}+\hat{j}+\hat{k}, \hat{i}-\hat{j}+\hat{k},-\hat{i}-\hat{j}+\hat{k}$ respectively is

Position vectors in 3D space:

Given vertices: $\vec{P} = -\hat{i} + \hat{j} + \hat{k}$, $\vec{Q} = \hat{i} + \hat{j} + \hat{k}$, $\vec{R} = \hat{i} - \hat{j} + \hat{k}$, $\vec{S} = -\hat{i} - \hat{j} + \hat{k}$

Adjacent sides from vertex P: $\vec{PQ} = \vec{Q} - \vec{P} = 2\hat{i}$, $\vec{PS} = \vec{S} - \vec{P} = -2\hat{j}$

These vectors are orthogonal with magnitudes $|\vec{PQ}| = 2$ and $|\vec{PS}| = 2$, confirming rectangular geometry.

The area follows from the cross product magnitude or product of adjacent sides:

$|\vec{PQ} \times \vec{PS}| = |(2\hat{i}) \times (-2\hat{j})| = 4$

$\boxed{4\text{ square units}}$