Tangent vector

Imagine you are taking a ride on a roller coaster. As you move along the track, you can feel yourself being pulled in different directions. The direction you're moving in at any given moment is called the tangent, and it's like a line that touches the track at that spot.

Now imagine you're moving along a curve, like a wiggly line on a piece of paper. At any point along the curve, you can draw a line that just touches the curve at that spot. That line is called a tangent too!

A tangent vector is kinda like an arrow that points in the direction of the tangent at a particular point on a curve. It tells you which direction you'll be moving if you keep going along the curve at that spot.

Even though it sounds complicated, a tangent vector is really just a way of describing how things move along curves. Think of it like a way of taking a picture of something moving along a bumpy road. The tangent vector is like a snapshot of the direction it's moving in at that exact moment.