13.3 13.1: SUMMARY

• A vector-valued function is a function of the form

  r(t) = 〈x(t), y(t), z(t)〉 = x(t)i + y(t)j + z(t)k

• We often think of t as time and r(t) as a “moving vector” whose terminal point traces out a path as a function of time. We refer to r(t) as a vector parametrization of the path, or simply as a “path.”

• The underlying curve traced by r(t) is the set of all points (x(t), y(t), z(t)) in R³ for t in the domain of r(t). A curve in R³ is also called a space curve.

• Every curve can be parametrized in infinitely many ways.

• The projection of r(t) onto the xy-plane is the curve traced by 〈x(t), y(t), 0〉. The projection onto the xz-plane is 〈x(t), 0, z(t)〉, and the projection onto the yz-plane is 〈0, y(t), z(t)〉.