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Mom's Basement University

Vector Functions and Space Curves

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll learn about vector-valued functions, which describe curves in 3D space. You’ll see how to differentiate and integrate them component by component.

The Concept

Section titled “The Concept”

A vector-valued function takes a parameter t and outputs a vector

r(t)=⟨x(t), y(t), z(t)⟩\mathbf{r}(t) = \langle x(t),\, y(t),\, z(t) \rangle

As t varies, the tip of the vector traces out a space curve. Each component is just an ordinary function of t.

Derivatives and Integrals

Section titled “Derivatives and Integrals”

The derivative is computed component by component

r′(t)=⟨x′(t), y′(t), z′(t)⟩\mathbf{r}'(t) = \langle x'(t),\, y'(t),\, z'(t) \rangle

This gives the tangent vector at each point. Its magnitude is the speed

speed=∣r′(t)∣=[x′(t)]2+[y′(t)]2+[z′(t)]2\text{speed} = |\mathbf{r}'(t)| = \sqrt{[x'(t)]^2 + [y'(t)]^2 + [z'(t)]^2}

Integration works the same way

∫r(t) dt=⟨∫x(t) dt,  ∫y(t) dt,  ∫z(t) dt⟩+C\int \mathbf{r}(t)\,dt = \left\langle \int x(t)\,dt,\; \int y(t)\,dt,\; \int z(t)\,dt \right\rangle + \mathbf{C}

Everything you know about single-variable derivatives and integrals applies here. You just do it three times, once per component.

Worked Examples

Section titled “Worked Examples”

Example 1: A helix

The vector function

r(t)=⟨cos⁡t, sin⁡t, t⟩\mathbf{r}(t) = \langle \cos t,\, \sin t,\, t \rangle

traces a helix. It circles around in x and y while climbing steadily in z.

The derivative (tangent/velocity vector) is

r′(t)=⟨−sin⁡t, cos⁡t, 1⟩\mathbf{r}'(t) = \langle -\sin t,\, \cos t,\, 1 \rangle

The speed is constant

∣r′(t)∣=sin⁡2t+cos⁡2t+1=2|\mathbf{r}'(t)| = \sqrt{\sin^2 t + \cos^2 t + 1} = \sqrt{2}

The blue curve is the helix spiraling upward. The orange arrow is the tangent vector at one point, showing the direction of motion. The green dot marks the point on the curve. As you rotate the view, you can see how the helix wraps around the vertical axis.

Example 2: Tangent line at a point

For the vector function

r(t)=⟨t2, t, t⟩\mathbf{r}(t) = \langle t^2,\, t,\, \sqrt{t} \rangle

find the tangent line at t = 1.

First, the point on the curve

r(1)=⟨1,1,1⟩\mathbf{r}(1) = \langle 1, 1, 1 \rangle

The derivative

r′(t)=⟨2t, 1, 12t⟩\mathbf{r}'(t) = \left\langle 2t,\, 1,\, \frac{1}{2\sqrt{t}} \right\rangle

At t = 1

r′(1)=⟨2,1,0.5⟩\mathbf{r}'(1) = \langle 2, 1, 0.5 \rangle

The tangent line is

L(s)=⟨1,1,1⟩+s⟨2,1,0.5⟩=⟨1+2s, 1+s, 1+0.5s⟩\mathbf{L}(s) = \langle 1, 1, 1 \rangle + s\langle 2, 1, 0.5 \rangle = \langle 1 + 2s,\, 1 + s,\, 1 + 0.5s \rangle

The blue curve is r(t) = (t², t, √t). The green dot marks the point (1, 1, 1) at t = 1. The orange arrow is the tangent vector r’(1) = (2, 1, 0.5), and the red line is the tangent line extending through that point in both directions.

Example 3: Integrating a vector function

∫⟨et, sin⁡t, t2⟩ dt=⟨et, −cos⁡t, t33⟩+C\int \langle e^t,\, \sin t,\, t^2 \rangle\,dt = \left\langle e^t,\, -\cos t,\, \frac{t^3}{3} \right\rangle + \mathbf{C}

Each component is integrated independently. The constant of integration is a constant vector (three constants, one per component).

Real-World Application

Section titled “Real-World Application”

Vector functions describe motion in 3D:

  • Game engines use vector functions for character paths, camera rails, and projectile trajectories
  • The derivative gives velocity (direction and speed), the second derivative gives acceleration
  • Bezier curves and splines in animation software are vector functions
  • Physics simulations track position, velocity, and acceleration as vector functions of time

Quiz

Section titled “Quiz”
The vector function $\mathbf{r}(t) = \langle \cos t, \sin t, t \rangle$ traces
The derivative $\mathbf{r}'(t)$ gives
The speed along a curve is
To integrate a vector function, you
The tangent line to $\mathbf{r}(t)$ at $t = a$ is