Why and how does negative velocity exist?

• Velocity is a vector. Speed is its magnitude.
• Position is a vector. Length (or distance) is its magnitude.

A vector points in a direction in space. A negative vector (or more precisely "the negative of a vector") simply points the opposite way.

If I drive from home to work (defining my positive direction), then my velocity is positive if I go to work, but negative when I go home from work. It is all about direction seen from how I defined my positive axis.

Consider an example where I end up further back than where I started. I must have had negative net velocity to end up going backwards (I end at a negative position). But only because backwards and forwards are clearly defined as the negative and positive directions, respectively, before I start.

So, does negative velocity exist? Well, since it is just a matter of words that describe the event, then yes. Negative velocity just means velocity in the opposite direction than what would be positive.

From the math point of view, you cannot have “negative velocity” in itself, only “negative velocity in a given direction”.

The velocity is a 3-dimension vector, there is no such thing as a positive or negative 3D vector.

However, if you consider the velocity in direction $\mathrm{x}$, where $\hat{\mathbf{e}}_{\mathrm{x}}$ is some unit vector giving a reference direction (say, "West"), then the velocity “in direction $\mathrm{x}$” is simply the scalar product of the velocity and $\hat{\mathbf{e}}_{\mathrm{x}}$. This quantity is a real number and can be negative. If it is negative, it is equal to $-1 \times \text{(velocity in direction -x)}$: compute the velocity in the opposite direction, and reverse the sign.

I think one of the main reasons that you have velocity is to isolate a particular direction of movement from your forward speed.

If you travel North north east, you can extract the speed at which you move eastwards by calculating your eastwards velocity (possibly 1/3 of your speed travelling NNE).

Negative velocities probably arrived as a consequence of the fact that when measuring a velocity, you have to define a direction.