Vector-Valued Functions

Definition

A vector-valued function $\mathbf{r}(t)$ maps a real number $t$ (often representing time) to a vector in space. It is the primary tool for describing curves and trajectories in multi-dimensional space.

• How to read: “Vector-valued function r of t.”
• Meaning: Input is a scalar parameter; output is a position vector tracing a curve through space.

Why It Matters

Objects don’t just exist at points; they follow paths through time. Vector functions map these paths. Without them, we couldn’t track an orbit or a missile trajectory, leaving us unable to predict where something will be based on where it was.

Core Concepts

• Component Form: $\mathbf{r}(t) = f(t)\mathbf{i} + g(t)\mathbf{j} + h(t)\mathbf{k}$, where $f, g,$ and $h$ are real-valued component functions.
  • How to read: “r of t equals f of t i plus g of t j plus h of t k.”
  • Meaning: Each component gives one coordinate as a function of $t$; differentiate or integrate component-wise.
• Continuity: A vector function $\mathbf{r}(t)$ is continuous at $t_0$ if and only if each of its component functions is continuous at $t_0$.
• Limits: $\lim_{t \to t_0} \mathbf{r}(t)$ is the vector whose components are the limits of $f, g,$ and $h$ as $t$ approaches $t_0$.
  • How to read: “Limit as t approaches t-zero of r of t.”
  • Meaning: Limits apply coordinate-by-coordinate; the curve has no jumps if all components are continuous.

Connected Concepts

• Derivative of Vector Functions
• Differentiation Rules for Vector Functions
• Integrals of Vector Functions