Tangent vector of a curve

Definition

Let $I \subseteq \mathbb{R}$ be an interval, let $\gamma \colon I \to \mathbb{R}^n$ be a parametrised curve, and let $t_0 \in I$. If the limit $$\gamma'(t_0) = \lim_{h \to 0} \frac{\gamma(t_0 + h) - \gamma(t_0)}{h} = \big( \gamma_1'(t_0), \dots, \gamma_n'(t_0) \big)$$ exists, it is the tangent vector, or velocity vector, of $\gamma$ at $t_0$. Its norm $\|\gamma'(t_0)\|$ is the speed of $\gamma$ at $t_0$. If $\gamma'(t_0) \ne 0$, the unit tangent vector is $T(t_0) = \gamma'(t_0) / \|\gamma'(t_0)\|$, and the tangent line at $\gamma(t_0)$ is $\{\gamma(t_0) + s \, \gamma'(t_0) : s \in \mathbb{R}\}$.

Remarks

At an end point of $I$, the limit is one-sided. For the circle $\gamma(t) = (\cos t, \sin t)$, $\gamma'(t) = (-\sin t, \cos t)$. It is orthogonal to $\gamma(t)$ and has length $1$. The tangent vector can be zero. For $\gamma(t) = (t^2, t^3)$, $\gamma'(0) = (0, 0)$, and the image has a cusp at the origin. The tangent vector depends on the parametrisation, but for a regular curve its direction does not change under an increasing change of parameter.

Explanations

L1 human

The tangent vector points in the direction in which the curve moves at time $t_0$. Its length tells how fast the point moves.