Whether points are necessarily "infinitesimally out in the tangent vector direction"

Question

In school I remember learning that the tangent vector points in the direction of the next point and a point is supposed to be infinitesimally out in the direction of the tangent vector, but now I double checked and the tangent definitely just touches the curves at a point, for instance the graph of $\langle t, t^{2} \rangle $ has a tangent curve $\langle 1, 2t \rangle $ so at the point $(0,0)$ the tangent line is below the curve after that point. So if there is supposed to be a point infinitesimally far out in the tangent direction for the example, what does that mean?

Answer 1

It's a handwavey way to make a mental connection between tangents and secants. But it is so handwavey that it's no surprise that you're confused by it. So first off, you are absolutely right that there doesn't need to be any more points on the tangent that are also on the curve it's tangent to, except of course the one point where it touches the curve.

Now here's what the description is supposed to make intuitive: The tangent line to a curve at a point $P$ is in some way the limit of the secant lines, where the secants intersect at $P$ and another point $S$, and $S$ tends to $P$. Here, the secant lines do point in the direction of an additional point on the line: $S$. However, there are always points on the curve between $P$ and $S$, so it's not the "next point" to $P$. The tangent is the limit of the secants as $S$ gets ever closer to $P$, so one might say that the tangent does point to the "next point" - if there were such a thing at all! But full disclosure, there is no "next point". No matter which point $N$ (for "next") we choose, there is always another point between $P$ and $N$, since the reals are dense (the rationals, too, by the way). But we can kind of imagine which line would point to that imaginary next point if it did exist, and that's the tangent.