---
title: "Vector-Valued Functions and Motion Along a Curve"
description: "Describe a path in space by a vector that depends on one parameter. Differentiating gives velocity and acceleration, and the arc length is an integral of speed."
canonical: https://lightmysky.com/learn/mathematics/vector-valued-functions-and-motion-along-a-curve-mt_Xcjt6exMy3
source: https://lightmysky.com/learn/mathematics/vector-valued-functions-and-motion-along-a-curve-mt_Xcjt6exMy3.md
retrieved: 2026-09-12
---

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# Vector-Valued Functions and Motion Along a Curve

Describe a path in space by a vector that depends on one parameter. Differentiating gives velocity and acceleration, and the arc length is an integral of speed.

Subject: Mathematics · Area: Calculus & Analysis · Ages 19 to 20
Page: https://lightmysky.com/learn/mathematics/vector-valued-functions-and-motion-along-a-curve-mt_Xcjt6exMy3

## Ready when they can

- Differentiate a vector-valued function componentwise to get velocity
- Find the unit tangent at a point on a space curve
- Set up the arc length of a curve as an integral of speed

## Lesson: Position, velocity, and the length of a path

A vector valued function takes one parameter and returns a position vector. As t moves, r(t) = (3t, 4t, 0) traces a line through space. Each component is an ordinary function of t, so your old calculus applies piece by piece.

**Example.** Differentiate each component to get velocity. For r(t) = (t, t squared), velocity is (1, 2t). Speed is the length of velocity: (3, 4, 0) has length 5, so that particle moves at speed 5.

The unit tangent is velocity divided by speed. It keeps the direction of motion and drops the size. For velocity (3, 4, 0) with speed 5, the unit tangent is (3/5, 4/5, 0).

**Tip.** Arc length is the integral of speed from a to b. When speed is constant, skip the integral: length is speed times time. Speed 5 from t = 0 to t = 2 gives length 10.

**Recap.** Differentiate component by component for velocity, divide by its length for direction, and integrate speed for the length of the path.

## Practice

16 questions on this page, each with its working shown.

## Needs first

- [Parametric Equations of Curves](https://lightmysky.com/learn/mathematics/parametric-equations-of-curves-mt_7NC7SCeU2P)
- [Integration by Substitution](https://lightmysky.com/learn/mathematics/integration-by-substitution-mt_jH-j_XsdC7)
- [Projectile Motion](https://lightmysky.com/learn/mathematics/projectile-motion-mt_O1C-QxCSJN)

## Opens up

- [Arc Length and Curvature of a Space Curve](https://lightmysky.com/learn/mathematics/arc-length-and-curvature-of-a-space-curve-mt_8mXYE8bVG1)
- [Functions of Several Variables and Level Curves](https://lightmysky.com/learn/mathematics/functions-of-several-variables-and-level-curves-mt_u04fZ_XiJ8)
- [Vector Fields and Line Integrals](https://lightmysky.com/learn/mathematics/vector-fields-and-line-integrals-mt_VZxKEmsVgO)