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.
What a learner can do afterwards
- 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
1 · Read
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.
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).
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.
Differentiate component by component for velocity, divide by its length for direction, and integrate speed for the length of the path.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.