Position, velocity, and acceleration in uniform circular motion

 Show: $\vec{r}$   $d\vec{r}$   $\vec{v}$   $d\vec{v}$   $\vec{a}$

As you start the motion, notice how vector $\vec{v}$ remains perpendicular to vector $\vec{r}$, at all times. This is because the magnitude of $\vec{r}$ remains constant, and only its direction changes. Thus the vector $d\vec{r}$ which describes the change in $\vec{r}$ ends up tangential to the circular trajectory (the $\vec{r}$-circle) in the limit of small angular increments.

The situation repeats itself for vector $\vec{a}$ which at all times remains perpendicular to the vector $\vec{v}$ (and along $d \vec{v}$, in the limit of small angular increments).

As a result, the vector $\vec{a}$ is always in the direction opposite to that of the vector $\vec{r}$, i.e. is directed toward the center of the circle; hence this acceleration is called centripetal.