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Mathematics LibreTexts

11: Vector-Valued Functions


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In the previous chapter, we learned about vectors and were introduced to the power of vectors within mathematics. In this chapter, we’ll build on this foundation to define functions whose input is a real number and whose output is a vector. We’ll see how to graph these functions and apply calculus techniques to analyze their behavior. Most importantly, we’ll see why we are interested in doing this: we’ll see beautiful applications to the study of moving objects.

  • 11.1: Vector–Valued Functions
    We are very familiar with real valued functions, that is, functions whose output is a real number. This section introduces vector–valued functions – functions whose output is a vector.
  • 11.2: Calculus and Vector-Valued Functions
    The previous section introduced us to a new mathematical object, the vector--valued function. We now apply calculus concepts to these functions. We start with the limit, then work our way through derivatives to integrals.
  • 11.3: The Calculus of Motion
    A common use of vector--valued functions is to describe the motion of an object in the plane or in space. A position function \(\vec r(t)\) gives the position of an object at time t . This section explores how derivatives and integrals are used to study the motion described by such a function.
  • 11.4: Unit Tangent and Normal Vectors
    Given a smooth vector-valued function r(t) , we defined that any vector parallel to r(t₀) is tangent to the graph of r(t) at t=t₀. It is often useful to consider just the direction of r⃗ ′(t) and not its magnitude. Therefore we are interested in the unit vector in the direction of r(t) . This leads to a definition of the unit tangent vector.
  • 11.5: The Arc Length Parameter and Curvature
    Approximation by multiple linear segments A curve in the plane can be approximated by connecting a finite number of points on the curve using line segments to create a polygonal path. Since it is straightforward to calculate the length of each linear segment, the total length of the approximation can be found by summing the lengths of each linear segment.


  • Gregory Hartman (Virginia Military Institute). Contributions were made by Troy Siemers and Dimplekumar Chalishajar of VMI and Brian Heinold of Mount Saint Mary's University. This content is copyrighted by a Creative Commons Attribution - Noncommercial (BY-NC) License.