Vector-Valued Functions

In this topic we define a vector-valued function in three dimensions.

Vector-valued functions are a natural extension of scalar functions and many tools from the calculus of one variable also extend to vector-valued functions.

We will illustrate how to evaluate a vector-valued function and determine its domain and range.

We also illustrate how to sketch a graph given an algebraic expression for a vector-valued function; and conversely how to determine a functional expression for a vector-valued function given information about its graph.

It is important to realize that there is not a one-to-one correspondence between a one-dimensional graph in 3D and a vector function. That is to say a vector-valued function has a graph, and only one set of points in 3D is it's graph. But a graph (set of points in 3D) can be represented by more than one functional rule. The classic 2D example is that of the unit circle. The unit circle can be parametrized as a vector-valued function by with ; and can also be parametrized as a vector-valued function by with . With both of these vector-valued functions we have the graph of the unit circle.
    It is this relationship between algebra and geometry that make vector-valued functions so important to study. For example, suppose a fly takes off from the top of a coffee cup at the front of a classroom. The path the fly traces as it travels through the classroom is a one-dimensional path which can be described as a vector-valued function. You may want to study the path (geometry only), but you may also want to know its speed, direction of motion, and acceleration at each point in time, in that case you will want more than its path -- you will want a vector-valued function that describes this fly's motion in three dimensions. At each point in time, you will no oonly have position but also its speed, direction of motion, and acceleration at each point in time.

Then, after introducing vector-valued functions, we show how to perform some basic operations like addition, subtraction, dot product and the cross product.

We will end this topic with a discuss of continuity followed with the graphs of more vector-valued functions.

(1) 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

    A vector-valued function can be extended defined with more than one variables an with more components. In general, a vector-valued function is a function that takes an -tuple of variables and output a unique vector with components. In this first example with take a random vector-valued function and explicitly say hat the scalar component functions are and ascertain the domain.

(2) 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:

    In the next example we notice that each scalar component function is linear and so claim know the vector-value function is in fact linear in 3D. Thus graphing this vector-value function is easy. Just pick any two point in the domain you wish, plot them and then draw a straight line.

(3) 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 ).

(4) 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.

(5) 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.

(6) 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.

(7) Definition (Operations with Vector-Valued Functions) Let and be vector-valued functions of the real variable , and let be a real-valued function. Then and are defined as follows:







These operations are defined on the intersection of the domain of the vector-valued and real-valued functions that occur in the definitions, respectively.

(8) Example (Limits of Vector-Valued Functions) Given and find

    Solution. We have,     

(9) Definition (Continuity of Vector-Valued Functions) A vector-valued function is continuous at means is in the domain of and . Further, a vector function is continuous on an interval if it is continuous at every point in the interval.

(10) Example (Continuity of Vector-Valued Functions) Determine where the vector-valued function is continuous.

    Solution. The function is continuous for all real numbers in its domain which is

(11) Example (Graphs of Vector-Valued Functions) Here are some graphs of vector-valued functions along with a random sampling of vectors in the range of   

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