The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages.


Vector Valued Functions

A vector valued function is a function F which maps real numbers to vectors. We shall study vector valued functions in either two or three dimensions. Here is the exact definition.

DEFINITION

A vector valued function in two dimensions is a set F of ordered pairs (t, X) such that for every real number t one of the following occurs.

(i) There is exactly one vector X in two dimensions for which the ordered pair (t, X) belongs to F. In this case F(t) is defined and F(t) = X.

(ii) There is no X for which (t, X) belongs to F. In this case F(t) is said to be undefined.

The definition of a three-dimensional vector valued function is similar. A vector valued function in two dimensions can be written as a sum

F(t) = f1(t)i + f2(t)j.

The functions f1 and f2 are real functions of one variable, called the components of F. The vector equation X = F(t) can also be written as a pair of parametric equations

x = f1(t), y = f2(t).

As t varies over the real numbers, the point X(x, y) traces out a parametric curve in the plane. The vector valued function F(t) is called the position vector of the curve.

The line with the vector equation X = P + tC is a parametric curve with position vector F(t) = P + tC and components

f1(t) = p1 + tc1, f2(t) = p2 + tc2.

Example 1: Movement Around the Unit Circle
Example 2: A Ball Thrown
Example 3: Cycloid
Example 4: Circular Helix
Example 5: Change of Price Vector


Last Update: 2006-11-25