Unit 9: Parametric Equations, Polar Coordinates & Vector-Valued Functions

Motion with Vector-Valued Functions

Analyzing position, velocity, and acceleration for motion along a curve.

Advanced25 min lesson2 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Derivatives of Parametric Functions

Position, Velocity, and Acceleration as Vectors

Definition — Vector-Valued Motion

A particle's position is r(t) = ⟨x(t), y(t)⟩. Velocity is v(t) = r'(t) = ⟨x'(t), y'(t)⟩, and acceleration is a(t) = v'(t) = ⟨x''(t), y''(t)⟩. Speed is the magnitude of velocity, |v(t)| = √([x'(t)]² + [y'(t)]²).

Worked Example — Finding Velocity, Acceleration, and Speed

A particle's position is r(t) = ⟨t², t³−3t⟩ — the same curve from the previous lesson. x(t)=t², x'(t)=2t, x''(t)=2. y(t)=t³−3t, y'(t)=3t²−3, y''(t)=6t. At t = 2: v(2) = ⟨4, 9⟩ (matching dx/dt=4 and dy/dt=9 found in the previous lesson at the same t — a direct cross-check). a(2) = ⟨2, 12⟩. Speed = |v(2)| = √(4²+9²) = √97 ≈ 9.849.

The velocity vector ⟨4, 9⟩ also carries a direction: both components positive place it in the first quadrant, with angle arctan(9/4) ≈ 66.0° above the positive x-axis — the direction the particle is instantaneously moving at t = 2.

Tip

Velocity is a vector — magnitude alone (speed) discards the direction of motion, and direction alone discards how fast it's happening. Both come from the same two derivatives.

Common Mistakes

Computing speed as x'(t) + y'(t) instead of √([x'(t)]² + [y'(t)]²).
Speed is a vector's magnitude, which requires the Pythagorean combination of its components, not their sum.

Key Takeaways

Velocity and acceleration for motion along a curve are vectors, built from the same component derivatives as parametric dy/dx.
Speed is the magnitude of the velocity vector; direction of motion comes from the velocity vector's components.

Summary

The final lessons of this unit turn to a third representation entirely — polar coordinates — and the area and derivatives that come with it.

Track your progress

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