Vector Functions and Space Curves

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.

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

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

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

The derivative is computed component by component

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

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

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

Integration works the same way

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

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

Example 1: A helix

The vector function

$$\mathbf{r}(t) = \langle \cos t,\, \sin t,\, t \rangle$$

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

The derivative (tangent/velocity vector) is

$$\mathbf{r}'(t) = \langle -\sin t,\, \cos t,\, 1 \rangle$$

The speed is constant

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

Drag to rotate · Scroll/Pinch to zoom

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

$$\mathbf{r}(t) = \langle t^2,\, t,\, \sqrt{t} \rangle$$

find the tangent line at t = 1.

First, the point on the curve

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

The derivative

$$\mathbf{r}'(t) = \left\langle 2t,\, 1,\, \frac{1}{2\sqrt{t}} \right\rangle$$

At t = 1

$$\mathbf{r}'(1) = \langle 2, 1, 0.5 \rangle$$

The tangent line is

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

Drag to rotate · Scroll/Pinch to zoom

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

$$\int \langle e^t,\, \sin t,\, t^2 \rangle\,dt = \left\langle e^t,\, -\cos t,\, \frac{t^3}{3} \right\rangle + \mathbf{C}$$

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

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