Space Curves and Vector-Valued Functions

has the unit circle as its graph, but these equations do not represent the same curve— because the circle is traced out in different ways on the graphs. Be sure you see the distinction between the vector-valued function and the real-valued functions and All are functions of the real variable but is a vector, whereas and are real numbers for each specific value of . Vector-valued functions serve dual roles in the representation of curves. By letting the parameter represent time, you can use a vector-valued function to represent motion along a curve. Or, in the more general case, you can use a vector-valued function to trace the graph of a curve. In either case, the terminal point of the position vector coincides with the point or on the curve given by the parametric equations, as shown in Figure 12.1. The arrowhead on the curve indicates the curve’s orientation by pointing in the direction of increasing values of t. x, y, z x, y r t t t h t g t , f t , r t t, h. g, f, r r sin t2 i cos t2 j r sin t i cos t j I. t h g, f, z h t y g t , x f t , f t , g t , h t C I. t g f y g t x f t f t , g t