Tangent Vector

Definition: Tangent Vector

Let $\gamma :I\subseteq \mathbb{R}\to {\mathbb{R}}^n$ be a parametric curve which is differentiable at $t\in I$.

The derivative $\dot{\gamma }(t)$ is known as the tangent vector of $\gamma$ at $t$.

NOTE

The tangent vector is also known as $\gamma$‘s velocity and its magnitude as $\gamma$‘s speed.

Definition: Unit Tangent Vector

The unit tangent vector is the unit vector obtained from the tangent vector:

$$\frac{1}{||\dot{\gamma }(t)||}\dot{\gamma }(t)$$

NOTATION

$$\mathbf{T}(t)$$