Vector-Valued Functions

Two separate equations, x(t) and y(t), tell you where a particle is, but they don't show it moving. Bundle them into one position vector r(t) = (x(t), y(t)) and now you have a single arrow from the origin to the particle, growing and swinging as t changes. That arrow is the whole path in one object: trace its tip and you get the curve; watch it tick forward in t and you get the motion.

Take the derivative of each coordinate separately: r'(t) = (x'(t), y'(t)). This is the velocity vector. At any instant, it points in the exact direction the particle is heading, because it's built from the instantaneous rates of change of x and y, the same way a tangent line shows the direction of a curve at a point. The position vector's length just tells you how far the particle is from the origin, which has nothing to do with how fast it's moving or which way. Speed only shows up when you take the magnitude of the velocity vector: speed = √((x'(t))² + (y'(t))²). That formula is really the distance formula applied to a rate instead of a displacement.

This topic is where parametric equations, which you already differentiate piece by piece, meet vectors as objects you can add and scale. It also sets up matrices later in the unit: a position vector can be multiplied by a matrix to rotate, stretch, or reflect the whole path, turning one moving point into a tool for describing much more complicated motion.