What is the connection between vector functions and space curves?

Question

I can't grasp what is the difference between vector functions and space carves. for example: $$\gamma(t)=(f(t),g(t),h(t))$$ I can assume this as a vector that starts from $(0,0,0)$ and points to a specific coordinate and also as a curve. In many cases they are both same but some times it must be clarified. for example when we define $T(t)$ as the tangent vector, what is tangent on? curve or vector?

Answer 1

This is a somehow conventional abuse of terminology. Let's say you have a map $$ \gamma:(0,1)\to \mathbb{R}^3 $$ usually with some extra properties such as differentiability. The image of $(0,1)$ under $\gamma$ is a "curve" in the space. Sometimes, people say that the map itself is the curve or a parametrization of the curve.

But there can be other map $\beta:(0,1)\to\mathbb{R}^3$ that shares the same image of $\gamma$.

The tangent vector at point $\gamma(t_0)$ is given by $\gamma'(t_0)$.

Hope this helps!