Introduction to Vector Functions

A vector-valued function, or simply a vector function, extends the concept of scalar functions by assigning a vector to each input value from its domain. In the context of motion through space, particularly in three dimensions, such functions are essential for describing trajectories and paths. A vector function r(t) is typically defined as:

\begin{equation*}\mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle\end{equation*}

Here, f(t), g(t), and h(t) are real-valued component functions that define the object's position along the x-, y-, and z-axes, respectively. The variable t often represents time, making r(t) a position vector describing motion in space.

As t varies over its domain, the position vector r(t) traces a space curve C, a path of a moving point in three-dimensional space. This path represents the collection of all terminal points P = (f(t), g(t), h(t)) reached by r(t). The domain of r(t) is determined by the set of values for which all three component functions are defined.

To study the behavior of vector functions near specific points, the concept of a limit is employed. The limit of a vector function as t→a exists if the limits of all its component functions exist:

\begin{equation*}\lim_{t \to a} \mathbf{r}(t) = \left\langle \lim_{t \to a} f(t), \lim_{t \to a} g(t), \lim_{t \to a} h(t) \right\rangle\end{equation*}

This formulation ensures that the vector approaches a definite value in both magnitude and direction.

A vector function r(t) is said to be continuous at a point t = a if:

\begin{equation*}\lim_{t \to a} \mathbf{r}(t) = \mathbf{r}(a)\end{equation*}

This implies that the function has no discontinuities at that point, and the path traced remains smooth and uninterrupted. The continuity of r(t) is directly dependent on the continuity of its component functions f(t), g(t), and h(t). If each of these functions is continuous at t = a, then r(t) is also continuous at that point.

These foundational concepts are critical for analyzing motion, especially in physics and engineering, where the spatial behavior of particles or objects over time must be precisely described and predicted.