Welcome to the dynamic world of Vector-Valued Functions. Unlike the static equations of the past, vector functions allow us to describe the trajectory of a moving point through space. Imagine a particle traveling through the void; its position at any given moment $t$ is defined by a vector anchored at the origin, pointing to its location in 3D space.
The Definition of a Space Curve
When we map a real-valued parameter $t$ to three separate component functions, we define a space curve $C$.
The set $C$ of all points $(x, y, z)$ in space, where: $$x = f(t) \quad y = g(t) \quad z = h(t)$$ and $t$ varies throughout an interval $I$, is called a space curve.
Alternatively, we use the vector notation: $$\mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle = f(t)\mathbf{i} + g(t)\mathbf{j} + h(t)\mathbf{k}$$ Here, $\mathbf{r}(t)$ is the position vector of a moving particle at time $t$.
Key Geometric Archetypes
- The Helix: A curve that spirals upward around a cylinder (usually $x^2 + y^2 = a^2$). This is the fundamental geometry of springs and the DNA double helix.
- The Twisted Cubic: A classic non-planar curve visualized as the intersection of two cylinders: $y = x^2$ and $z = x^3$. It warps through all three dimensions simultaneously.
Examples from the Field
Describe the curve defined by $\mathbf{r}(t) = \langle 1 + t, 2 + 5t, -1 + 6t \rangle$.
Analysis: This is a parametric equation for a straight line. It passes through the point $(1, 2, -1)$ and follows the direction vector $\mathbf{v} = \langle 1, 5, 6 \rangle$.
Sketch the curve $\mathbf{r}(t) = \cos t \mathbf{i} + \sin t \mathbf{j} + t \mathbf{k}$.
Analysis: The components $x = \cos t$ and $y = \sin t$ satisfy $x^2 + y^2 = 1$, meaning the curve stays on a circular cylinder. As $t$ increases, $z=t$ pulls the point upward, creating a spiral.
Using a computer to visualize $\mathbf{r}(t) = \langle t, t^2, t^3 \rangle$.
Analysis: This curve is "twisted" because it is the intersection of the parabolic cylinder $y = x^2$ and the cubic cylinder $z = x^3$. It is a standard example of a curve that does not lie in a single plane.