Substituting the circumference formula for "X" and the symbol for the period of time makes the formula

since the velocity in all of these formulas is the magnitude only and the magnitude of the velocity never changes.

therefore

To define a velocity formula, we need to look at the definition of a vector. But first a question.

Apply our vector knowledge to circular motion. To begin with we need some velocity vectors. Watch the animation below to see where we will get 8 velocity vectors.

We know that two vectors are the same if they have the same magnitude and direction.(This is what your answer to the question above illustrates.) This means that if I slide a vector anywhere on the page and it will be the same vector as long as it has the same direction and magnitude. Watch the animation below to how we are finding 8 velocity vectors for 8 different positions.

Look for a pattern to derive a new formula. On the circle on the left the circumference represents the change in position. Because circle is traveled once around the time is the period. This gives us the formula:

The circle on the right replaces the radius vector with the velocity vector. This means that we can make the same substitution in the formula above.

What is defined as the change in velocity over the change in time?