2.1 Vector-Valued Functions and Space Curves

Vector-Valued Functions

A vector-valued function assigns a vector to each input value, rather than a single number. This lets you describe motion and paths through 3D space with a single function. Understanding these functions is the foundation for everything else in this unit: velocity, acceleration, curvature, and arc length all build directly on what's covered here.

Concept of Vector-Valued Functions

A vector-valued function maps a real number $t$ (the parameter) to a vector in space:

$\mathbf{r}(t) = \langle f(t),\, g(t),\, h(t) \rangle$

Each component is just an ordinary scalar function:

• $f(t)$ controls the $x$-coordinate
• $g(t)$ controls the $y$-coordinate
• $h(t)$ controls the $z$-coordinate

As $t$ changes, the tip of the vector $\mathbf{r}(t)$ traces out a curve in space. Think of $t$ as time: at each moment, the function tells you a position $(x, y, z)$.

You can also write the same function using standard basis vectors:

$\mathbf{r}(t) = f(t)\,\mathbf{i} + g(t)\,\mathbf{j} + h(t)\,\mathbf{k}$

This is equivalent to the angle-bracket notation; which one you use is just a matter of preference.

Representation of Space Curves

A space curve is the path traced by $\mathbf{r}(t)$ as $t$ varies over some interval. The corresponding parametric equations are:

$x = f(t), \quad y = g(t), \quad z = h(t)$

Two common examples worth knowing well:

• Circular helix: $\mathbf{r}(t) = \langle \cos t,\, \sin t,\, t \rangle$. The $x$ and $y$ components trace a circle while $z = t$ steadily increases, so the curve spirals upward around the $z$-axis. The radius is 1, and the helix rises by $2\pi$ units for each full revolution.
• Unit circle in the $xy$-plane: $\mathbf{r}(t) = \langle \cos t,\, \sin t,\, 0 \rangle$. With $z$ fixed at 0, this is a planar curve sitting in 3D space. It's periodic with period $2\pi$.

Notice how changing even one component (replacing $z = 0$ with $z = t$) transforms a flat circle into a helix. That's the power of the parametric setup.

Graphing and Analysis

Graphs of Vector-Valued Functions

To sketch a space curve by hand:

1. Build a table of values. Pick several $t$ values (e.g., $t = 0, \frac{\pi}{4}, \frac{\pi}{2}, \pi, \ldots$) and compute $\mathbf{r}(t)$ for each.
2. Plot the points $(f(t),\, g(t),\, h(t))$ in 3D coordinates.
3. Mark the orientation. Add arrows showing the direction of increasing $t$. This matters because two curves can trace the same path in opposite directions.
4. Connect the points with a smooth curve, following the orientation.

A few special cases to recognize:

• Planar curves stay in a single plane. For example, $\mathbf{r}(t) = \langle 2\cos t,\, 3\sin t,\, 1 \rangle$ traces an ellipse in the plane $z = 1$.
• Closed curves return to their starting point. If $\mathbf{r}(a) = \mathbf{r}(b)$ for the endpoints of your domain, the curve closes up.

For anything beyond simple curves, graphing software (Desmos 3D, GeoGebra, Mathematica) is extremely helpful for building intuition about what these curves look like.

Domain and Range of Vector Functions

The domain of $\mathbf{r}(t)$ is the set of all $t$ values where every component function is defined. To find it:

1. Determine the domain of each component $f(t)$, $g(t)$, and $h(t)$ individually.
2. Take the intersection of those domains. Every component must be defined simultaneously.

For example, if $\mathbf{r}(t) = \langle \sqrt{t},\, \ln t,\, t^2 \rangle$, then $\sqrt{t}$ requires $t \geq 0$, $\ln t$ requires $t > 0$, and $t^2$ is defined everywhere. The intersection gives a domain of $t > 0$.

The range is the set of all output vectors, which geometrically is the set of points the curve passes through. The character of the range depends on the component functions:

• Periodic components (like $\sin t$, $\cos t$) create repeating or closed patterns.
• Bounded components keep the curve within a finite region. The unit circle has range confined to vectors of length 1 in the $xy$-plane.
• Unbounded components (like $z = t$ in the helix) cause the curve to extend infinitely.

In applied problems, the domain is often further restricted by physical context. If $t$ represents time starting from launch, you'd restrict to $t \geq 0$ even if the components are defined for negative $t$.