In uniform circular motion, the acceleration $\vec{a}$ is purely centripetal. It points from the particle's position $\vec{r}$ directly towards the center of the circle $\vec{r}_c$. The magnitude of the centripetal acceleration is given by:

$$a = \frac{v^2}{R}$$

where $v = |\vec{v}|$ is the particle's speed and $R$ is the radius of the circular path.

The vector from the particle to the center of the circle, $\vec{R} = \vec{r}_c - \vec{r}$, has magnitude $R$ and points in the same direction as the acceleration $\vec{a}$. We can write this vector as:

$$\vec{R} = R \hat{a} = R \frac{\vec{a}}{a}$$

The position of the center is therefore $\vec{r}_c = \vec{r} + \vec{R} = \vec{r} + R \frac{\vec{a}}{a}$.

From the centripetal acceleration formula, the radius is $R = v^2/a$. Substituting this into the expression for $\vec{r}_c$:

$$\vec{r}_c = \vec{r} + \left(\frac{v^2}{a}\right) \frac{\vec{a}}{a} = \vec{r} + \frac{v^2}{a^2}\vec{a}$$

Since $v^2 = |\vec{v}|^2 = \vec{v} \cdot \vec{v}$ and $a^2 = |\vec{a}|^2 = \vec{a} \cdot \vec{a}$, we can express the final result in vector form.