Vector Functions

    In this topic, we plan to study vector-valued functions (vector functions) by evaluating, sketching some graphs, taking some limits, and most importantly exploring the notion of continuity of a vector function. We start of by defining a vector function of one variable and then give several examples.

    Vector functions are useful for tracing out graphs of curves and for describing motion along a path. Often the variable represents time and since each represents a vector, we have a position at time That is to say, given a time value of we have a vector which represents a point where and In this manner, we use arrowheads on the curve to indicate the curve's orientation by pointing in the direction of increasing values of

Definition (Vector-Valued Function) A vector-valued function of a real variable with domain assigns to each number in the set a unique vector . The set of all vectors of the form for in is the range of . In three dimensions vector functions can be expressed in the form,
  


where are real-valued functions of the real variable defined on the domain set . A vector function may also be denoted by Unless stated otherwise, the domain of a vector function is the intersection of the domains of the component functions and

Example (Evaluate a Vector Function) If then the component functions are

   and    

where The domain of is the interval [0,3) since requires and requires We can evaluate the vector function at as follows:

    Sketching the graphs of vector functions can be challenging. The idea is to give the illusion of a three-dimensional drawing on a two-dimensional piece of paper.

Example (Sketch a Vector-Valued Function) Sketch the graph of the vector-valued function


   
    Solution. The graph is the set of all points with    and   The graph is a line that passes through (when ) and (when ).

Example (Sketch a Vector-Valued Function) Sketch the graph of the vector-valued function

  

    Solution. The graph is the set of all points with   and The graph is a circular helix that lies on the surface of the cylinder with equation The cylinder is centered at (0,0) in the -plane.

Example (Sketch a Vector-Valued Function) Sketch the graph of the vector-valued function



    Solution. The graph is the set of all points with    The curve has the shape of a parabola and passes through the points (when ) and when which is the vertex.

Example (Domain of a Vector-Valued Function) Find the domain of the vector-valued function given by

  

    Solution. The domain is   , since demands that The graph is all with and To help sketch a graph we can eliminate by using and then . Solving for , which can be used to help make a 3-D graph.

Example (Find a Vector-Valued Function)  Find a vector-valued function whose graph is the curve of intersection of the hemisphere   and the parabolic cylinder .

    Solution. One way to accomplish the task is by letting Then and   Therefore a value-valued function for this intersection is     which has the following graph.



Example (Find a Vector-Valued Function) Find a vector-valued function whose graph is the curve of intersection of the plane and the plane

    Solution. One way to accomplish the task is by letting Then to find relations for and we will solve the system

.

Eliminating we have, and so   Solving the first for we find



Therefore a vector-valued function for this intersection is   which has the following graph.

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Cite this as:
Vector Functions
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/vector-functions.html