 Hello and welcome to this screencast on Section 9.6, Vector Value Functions. Consider a curve in three space. You can think of a point on this curve as resulting from a vector from the origin to the point. As the point travels along the curve, the vector changes in order to terminate at the desired point. The terminal points of the vector trace out the curve in space. From this perspective, the x, y, and z coordinates of the point are functions of time t, like we saw in the last section for the parametric equations of a line. A vector-valued function is a function whose input is a real parameter t and whose output is a vector that depends on t. The graph of a vector-valued function is the set of all terminal points of the output vectors with their initial points at the origin. The parametric equations for a curve are equations that describe the x, y, and z coordinates of a point on the curve. Note, particularly, that every set of parametric equations determines a vector-valued function of the form given here. As an example of a vector-valued function, the curve we looked at earlier has x component given by cosine of t, y component given by sine of t, and z component given by cosine of t times sine of t.