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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.