Vector Functions Summary

Calculus 3 · Axiom Academy

How a moving vector traces a curve in space, and how its calculus reveals velocity, acceleration, arc length, and curvature. One vector, one curve. A vector function traces a space curve; its limits, continuity, derivatives, and integrals are all handled component-wise . Geometry falls out of calculus. The first derivative is the velocity — tangent to the curve; the second derivative is the acceleration. Arc length is the integral of speed. measures the true distance travelled along the winding path. Curvature measures bending. — large for tight turns, zero for a straight line. The TNB frame rides along. form a moving right-handed frame that splits acceleration into a speeding-up part and a turning part. Core Concept Vector-Valued Functions A vector function maps each real number t to a position vector, tracing a parametric curve through space as t varies. Lines, helices, circular paths, and projectile arcs are all vector functions. Limits & continuity: evaluated component-wise; continuous when every component is. Watch out for: the curve is the image, not the formula — different parametrizations can trace the same path. Core Concept Velocity & Acceleration Differentiate component-wise. The velocity is tangent to the curve and points in the direction of motion; the acceleration measures how velocity changes. Speed: — a scalar, how fast position changes. Watch out for: constant speed does not mean zero acceleration (a turn is acceleration too).

This is the written version of the interactive lesson above. See the full Calculus 3 course.