Position, velocity, and acceleration in uniform circular motion

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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.