Vector Functions

In calculus of a single variable one often thinks of a function x=f(t) as representing the position of a particle on a line. This notion can be extended to more than 1 dimension. Two functions are required to describe the position of particle in two dimensions. In three dimensions, 3 functions are required.

Consider the following two-dimensional vector function:

displaymath27

The x component of r is 2cos(t) and y component of r is sin(t). Hence, we can also describe the vector function by writing x(t)=2cos(t) and y(t)=sin(t).

For each t, r(t) corresponds to a point in the xy plane. We graph r(t) by plotting these points for 0<=t<=2*pi, as shown below.

The model vector function <2cos(t),sin(t)> traces out an ellipse. Since x=2cos(t) and y=sin(t), we have:

displaymath29

If we think of r(t) as representing the position of a particle then r(1)=<2cos(1),sin(1)>. r(t) can also be thought of as vector. The vector in the plot is r(1), with its tail starting at the origin.

In addition to position functions of particles, vector functions also describe space curves. In our example above, the space curve is an ellipse. A key point is that there a several vector functions that represent the same ellipse. For example,

displaymath31

traces out precisely the same ellipse as the model function above. However the two vector functions correspond to different position functions. In the first case, the particle requires 2*pi time units to get back to its starting point. For the second function, only pi time units are required.

Here is an example of a three-dimensional vector function:

displaymath33

which is plotted below for 0<=t<=7*pi. The space curve generated by this vector function is called a circular helix.

Vector functions are vectors and obey rules of addition and scalar multiplication. One can also compute the dot product and cross product of two vector functions.


[Vector Calculus Home] [Math 254 Home] [Math 255 Home] [Notation] [References]

Copyright © 1996 Department of Mathematics, Oregon State University

If you have questions or comments, don't hestitate to contact us.