How to evaluate vector-valued functions

 Goal

 Evaluate vector valued functions

Definition

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

The functions f, g, h are real-valued functions of the parameter t. Vector-valued functions are also written in the form:

The first form defines a two-dimensional vector-valued function. The second form describes a tri-dimensional vector-valued function. 

Example

For each of the vector-valued functions, evaluate r(0), r(π/2), r(2π/3). Do any of these functions have domain restrictions?

Solution

a. Let's substitute each of the value of t in the function:

To determine a domain restriction let's consider each component function separately. The function cost is defined for all values of t. The function sint is also defined for all values of t. Therefore the function r(t) is defined for all values of t

b. Let's do the same thing for the second function:

 

The component functions tant and sect are not defined for odd multiples of π/2. Therefore the vector valued-function r(t) is not defined for odd multiples of  π/2. 

Practice

For the vector-valued function r(t) = (t²-3t)i + (4t + 1)j, evaluate r(0), r(1), r(-4). Does this function have any domain restrictions?