Tangent Lines and Tangent Vector Fields
Definition
Let there be a given regular curve $\alpha (t)$.
- The vector field $\displaystyle T(t) := {{ d \alpha / d t } \over { \left| d \alpha / d t \right| }}$ is called the Tangent Vector Field.
- The line $l$ defined as follows in $t = t_{0}$ to $\alpha$ is called the Tangent Line. $$ l := \left\{ \mathbf{w} \in \mathbb{R}^{3} : \mathbf{w} = \alpha \left( t_{0} \right) + \lambda T \left( t_{0} \right) , \lambda \in \mathbb{R} \right\} $$
Explanation
The tangent vector field is an extremely important vector function in differential geometry, considering the direction of the tangent to the regular curve while standardizing its magnitude to $1$. It represents only the direction regardless of how sharply the curve bends.