In the cube of side 'a' shown in the figure, the vector from the central point of the face ABOD to the central point of the face BEFO will be:

Let the central point of face ABOD be $$G$$ and the central point of face BEFO be $$H$$.

Coordinates of vertices from the figure:

$$\text{O} = (0, 0, 0)$$, $$\text{D} = (a, 0, 0), \quad \text{B} = (0, 0, a), \quad \text{A} = (a, 0, a)$$, $$\text{F} = (0, a, 0), \quad \text{E} = (0, a, a)$$

Face ABOD lies in the $$xz$$-plane ($$y = 0$$): $$\vec{r}_G = \left(\frac{a}{2}\right)\hat{i} + 0\hat{j} + \left(\frac{a}{2}\right)\hat{k}$$

Face BEFO lies in the $$yz$$-plane ($$x = 0$$): $$\vec{r}_H = 0\hat{i} + \left(\frac{a}{2}\right)\hat{j} + \left(\frac{a}{2}\right)\hat{k}$$

Vector from $$G$$ to $$H$$: $$\vec{r}_{GH} = \vec{r}_H - \vec{r}_G$$

$$\vec{r}_{GH} = \left[0\hat{i} + \frac{a}{2}\hat{j} + \frac{a}{2}\hat{k}\right] - \left[\frac{a}{2}\hat{i} + 0\hat{j} + \frac{a}{2}\hat{k}\right] = -\frac{a}{2}\hat{i} + \frac{a}{2}\hat{j} = \frac{1}{2}a(\hat{j} - \hat{i})$$