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:

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:


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,

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:

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.