Now that we know what vectors in a plane and in space are about and how to represent them we are able to "invent" a function called "vector function". In this function we will map a real number to a vector; the vector will either be a plane vector or a space vector. The graph of such a function will be a referred to as a "space curve" and the graph of a vector function will also be the graph of the parametric equations that are given as the coordinate functions of the vector function.
This is an overview; now the details follow from our text:
See 12.1 Space curve and Vector-Valued Function
The next example illustrates a very important connection you want to make between parametric equations and a vecto-valued function. The idea is this: "When you graph a vector, say r(t) = (2t + 1) i + (3t +2) j you get an arrow tipped line that has the 2t + 1 for the X- coordinate and the 3t + 2 for the Y- coordinate. this means that graphing the parametric equations x = 2t + 1 and y = 3t + 2 "generates" the same points as does a plot of the tip of the vector r(t) = (2t + 1) i + (3t + 2) j . When graphing a vector valued function, think of following the tip of an arrow tipped line as its direction and magnitude change. You "see" this in this next example.
In the next example (helix) the arrow tipped vector for the vector valued function is not shown; this is customary. I ask that you try to visualize arrow tipped vectors from the origin to the points on the circular helix curve as it "climbs" the side of the right circular cylinder.
The next example illustrates that a curve can be given and then a vector function representation can be written for that curve. Notice that the representation is not unique just as parametric representations for curves are not unique.
The final example illustrate the same idea as the previous but is in space rather than the plane.
We could now spend time studying limit and continuity for vector functions. We would find that no new information beyond that studied in a first Calculus course is introduced and that the limit and the continuity for the vector function are simply done on the component functions which are real variable functions from Precalculus. It is crucial that we know that the limit of a vector function is done by finding limits of the coordinate functions because this will allow for differentiation and integration of the vector functions which is the topic we study next. The definition for limit would appear as follows:
Look at the following solved problems
in the Exercises at the end of
12.1 : 9,13,17,19, 23,25,27,39,41
Then do the following unsolved
problems as homework: 10,14,18,20,24,26,28,40