{wiki}

A <tangent vector> at a point $p$ of a <smooth manifold> without boundary records an instantaneous <velocity> through $p$. Its base point is part of the data; measuring its length additionally requires a <Riemannian metric>. Intrinsically, a <smooth curve> $\gamma$ with $\gamma(0)=p$ defines its <tangent vector> by how every local <smooth function> changes along it:
$$
v(f)=\left.\frac{d}{dt}f(\gamma(t))\right|_{t=0}.
$$
This is a <derivation at a point>. Two <smooth curves> through $p$ represent the same <tangent vector> if these <derivatives> agree for every such <smooth function>; equivalently, their coordinate <velocity vectors> agree in one <manifold chart>. The <chain rule> makes that agreement independent of the chosen <manifold chart>. https://www.math.toronto.edu/~jeffrey/matd67/tu.pdf[Loring W. Tu, An Introduction to Manifolds, §8.6: curves and intrinsic velocity]{ref}

For an <embedded submanifold> of <Euclidean space>, the intrinsic <tangent vector> is represented by the ordinary <derivative> $\gamma'(0)$, with its base point $p$ retained. A drawn arrow starting at $p$ represents that <tangent vector>; its endpoint $p+\gamma'(0)$ need not lie on the <embedded submanifold>. Nor is a <tangent vector> required to be a <unit tangent vector>: a stationary <smooth curve> gives the <zero vector>, which belongs to the <tangent space> but has no direction to normalize.

Here are explicit calculations. In $\mathbb R^2$, $\gamma(t)=(t,t^2)$ and $\eta(t)=(t,2t^2)$ have different routes but the same <tangent vector> $(1,0)$ at $(0,0)$. For $f(x,y)=3x+y$, both give $v(f)=3$. For a <circle> of <radius> $R$, $c(t)=(R\cos(\omega t),R\sin(\omega t))$ has <tangent vector> $(0,R\omega)$ at $(R,0)$: changing $\omega$ changes the <speed> or reverses travel without changing the <circle>.

A <reparametrization> $\beta(s)=\gamma(h(s))$ with $h(0)=0$ gives, by the <chain rule>, $\beta'(0)=h'(0)\gamma'(0)$. A regular change of <curve parameter> has $h'(0)\ne0$. The singular choice $h(s)=s^3$ instead gives
$$
\gamma(t)=(t,t^2),\qquad \beta(s)=(s^3,s^6),\qquad \gamma'(0)=(1,0),\qquad \beta'(0)=(0,0).
$$
Both describe the same <parabola>, whose <tangent line> at the origin is still the horizontal axis. The zero <velocity> diagnoses this singular change of <curve parameter>, whose inverse is not differentiable at zero; it does not show that the geometric <tangent line> is absent. The <regular curve> $\gamma$ supplies its nonzero direction.

\Image[/tangent-vectors-spaces-planes-curve-velocities.png]
{title=One parabola, different velocities: a singular parameter change can stop the arrow}

The diagram is an analytical illustration of these exact <parametric curves>, with equal parameter increments marked on each route. It separates geometry from traversal: the <tangent line> depends on the route, while a <velocity vector> also depends on how the <curve parameter> advances.


 Tangent vector (source code)