Mathematical Structure
Vector-Valued Functions

Prerequisites:

• Groups
• Vector Spaces

Ordinary functions are used for many things -- for example, when an object is dropped from a height of five feet its height t seconds after it is dropped is given by the function

Because the height is described by a single number we use a function whose value at each time is a number. When we describe the travels of a baseball in two dimensions we have two choices.

• We can use two functions x(t) and y(t) that describe its coordinate on the x-axis at time t and its coordinate on the y-axis at time t respectively. ... or ...

• We can use a single vector-valued function V(t) = (x(t), y(t)) that describes its location in R^2 at time t.

Definition:

A vector-valued function is a function F(t) whose values are vectors in some vector space V.

Examples

• Suppose that a person walks from the point U = (1, 3) to the point V = (6, 7) in a straight line at a steady speed starting at time t = 0 and reaching V at time t = 1. His location at time t can be described by the vector-valued function
  P(t) = (1 - t) U + t V

• Suppose that a rocket ship is coasting in a straight line from the point U to the point V starting at U at time t = 0 and reaching V at time t = 1. His location at time t can be described by the vector-valued function
  P(t) = (1 - t) U + t V

The CAS files below demonstrate how each computer algebra system works with vector-valued functions with values in R^n.