The definitions seem exactly the same to me. We have a(t) to be a vector function and x(t) to be a parametric curve where the tip of the position vector traces out a curve in an $n$-dimensional space as the input varies throughout it's domain values of t.

Vector Functions:

Vector functions are given by a: I $\rightarrow\mathbb{R}^n$ with the domain I, an interval of $\mathbb{R}$, and the range being a subset of two or three dimensional space. For example, with $n = 3$, with respect to the fixed orthonormal basis ${\bf{e}}_1$, ${\bf{e}}_2$ and ${\bf{e}}_3$. The vector ${\bf{a}}(t)$ can be written as $${\bf{a}}(t)=a_1(t){\bf{e}}_1 + a_2(t){\bf{e}}_2 + a_3(t){\bf{e}}_3.$$

Parametric Curves:

A parametric curve is a function x : I$\mathbb{R}^n$, x : t $\mapsto$ x(t) where I is an interval. We will consider $n = 2$ (two-dimesional parametric curves, or plane curves) and $n = 3$ (three-dimensional parametric curves, space curves). Some definitions further require the function x to be differentiable. As the parameter t varies in the interval I, the point with position vector x(t) traces out a curve.

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    $\begingroup$ What's "an interval of $\mathbb R^n$"? $\endgroup$ – Calvin Khor Jan 12 at 14:09
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    $\begingroup$ That was an error. I've corrected it to $\mathbb{R}$. Thanks. $\endgroup$ – user503154 Jan 12 at 14:17
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    $\begingroup$ In which case it looks the same to me, except that curves usually have some amount of smoothness to them. Maps $\mathbb R^n \to \mathbb R^m$ (which is what I previously thought you called vector functions) are of course not the same $\endgroup$ – Calvin Khor Jan 12 at 14:26

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