Introduction to Vector Functions

Calculus 3 · Axiom Academy

LESSON Introduction to Vector Functions Exploring parametric curves in space through vector-valued functions that trace paths as parameters change Each component function x(t) , y(t) , and z(t) is a scalar function of the parameter t . As t varies, the output vector traces out a curve in space — and the tip of that arrow is the point doing the moving. Let's visualize how a vector function traces a curve. Consider the simple 2D example: As t varies from 0 to , this traces a circle of radius 2 centered at the origin. Watch how the parameter t determines the position along the curve: Vector functions truly shine in 3D. Consider a helix : The x and y components create circular motion in the xy -plane, while the z component increases linearly, creating a spiral-staircase effect in 3D space. This could represent the path of a particle moving in a helical pattern, like a spring or a DNA strand. Just like scalar functions, vector functions have domains and ranges: The domain is restricted by the component functions. For example, if , then . The range is the actual curve traced in space. 5. Continuity of Vector Functions A vector function is continuous at t = a if: Equivalently, is continuous if and only if each component function x(t) , y(t) , and z(t) is continuous. This means the curve has no breaks or jumps. Continuous curves represent smooth, physically realizable paths. Discontinuous curves have sudden jumps — like a teleporting particle.

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