

Vector Functions and their Graphs
Purpose
The purpose of this lab is to give you practice using Maple to plot
and animate graphs of vector functions, to compute curvature, and to
specify the graphs' shapes.
Background
As we've seen in class, computing the curvature, , and the
unit normal and tangent vectors is a tedious process, even for the
simplest of curves. Fortunately, Maple procedures can be written to do
these calculations and this lab will introduce you to the ones that we
have written here at WPI as part of the CalcP package.
Before you can use any of these commands, you must load the package with the
following command. Since some of the commands use the linalg
package, it is probably a good idea to load it as well.
> with(linalg):
Warning, new definition for norm
Warning, new definition for trace
> with(CalcP):
The list below gives the names of the procedures we will be using as well
as brief descriptions of what each does. Maple help screens are available for all
of these procedures, so refer to them for further examples.
 VDiff
 Differentiates vectorvalued functions
 VPlot
 Plots vectorvalued functions in two and three dimensions
 VMag
 Computes the magnitude of a vector
 ParamPlot
 Animates parametric curves in two dimensions
 ParamPlot3D
 Animates parametric curves in three dimensions
 Speed
 Computes the speed of a particle moving on a path defined by a vectorvalued function r(t)
 unitvect
 Computes the unit vector associated with a vector v
 tanvect
 Computes the unit tangent vector, T(t), for a vectorvalued function r(t)
 normalvect
 Computes the unit normal vector, N(t), for a vectorvalued function r(t)
 Curvature
 Computes the curvature, (t), for a vectorvalued function r(t)
Examples
For more examples, see the help screens for the individual procedures.
Note that all of the procedures
accept either Maple vectors or Maple lists as arguments. Note also
that several of the procedures allow you to evaluate the result at a
specific value of t by using a second argument of the form
t=a (to evaluate at t=a). If the second argument is simply
t, then the result of the procedure is an expression involving
t. This does not apply to the VDiff command, however, where
argumets after the first are used to indicate derivatives.
> circ := t > vector([12*sin(t),12*cos(t)]);
> VDiff(circ(t),t);
> tanvect(circ(t),t);
> normalvect(circ(t),t);
> Curvature(circ(t),t);
> r := t > vector([t,t^2]);
> tanvect(r(t),t);
> tanvect(r(t),t=0);
> tanvect(r(t),t=1);
Computing the unit normal vector is always more
complicated than computing the unit tangent vector . In
addition, Maple likes to do computations as generally as possible,
which can cause complications. In particular, Maple assumes that all
variables can be complex numbers. This isn't a good assumption in
calculus, so the tanvect, normalvect, and Curvature
procedures had to be written to give results that are real numbers. A
sideeffect of this is that
results of the commands, especially the normalvect command, do
not always appear in the simplest form. Consider the following example.
> normalvect(r(t),t);
The notation appears because Maple
had to differentiate the absolute value function to obtain the normal
vector. Recall that the deriviative of is 1 if x > 0
and 1 if x < 0. Maple's notation stands for the derivative of , evaluated
at . Because Maple assumes that t can be complex, it allows for
to be negative. In our calculations, can never
be negative so the value of is
simply 1.
This may seem like a pain, but it won't affect your ability to compute
normal vectors at fixed values of t, as shown in the following
examples, or plot normal vectors. Putting the absolute value in several
of the CalcP package functions for curve comptations was a necessary
evil, because leaving it out produced answers that were just plain
wrong. By putting it in, the procedures give the correct answer, but
in a form that is more complicated than we would like.
> normalvect(r(t),t=0);
> normalvect(r(t),t=1);
> Curvature(r(t),t);
The absolute value also appears in the output of the Curvature
procedure, as shown above. This means that if you differentiate the
output of the Curvature function, for example to find extreme
values, you may see the Maple abs procedure in the results. You
shouldn't be alarmed by this, just remember that the value will always
be 1. Fortunately, the Maple solve command usually handles
cases involving the abs procedure just fine.
In the following example, plots of a vectorvalued function and its
curvature are generated. You should compare the two plots and try to
understand their relationships.
> plot(Curvature(r(t),t),t=2..2);
> VPlot(r(t),t=2..2);
The next few commands deal with a simple example of a curve known as a
helix.
> h := t > vector([cos(t),sin(t),t]);
> VPlot(h(t),t=0..4*Pi);
> ParamPlot3D(h(t),t=0..4*Pi);
> tanvect(h(t),t);
> Speed(h(t),t);
> Curvature(h(t),t);
> normalvect(h(t),t);
More Examples
These commands will show you how to find a line tangent to a vector valued
function at a given point, then graph both on the same plot.
> with(CalcP): with(linalg):
> circ := t > vector([12*sin(t), 12*cos(t)]);
You may use options to include more information on your plots, or to make
it easier to read. Use accents (not single quotes) to designate a string
for a title.
> VPlot(circ(t), t=0..2*Pi, color=RED, axes=NORMAL,
labels=[x,y], title=`circ(t)`);
This command will find the tangent vector at a given point, but it will
not give you a tangent line.
> slope:= tanvect(circ(t),t=Pi/3);
Maple's tangentline command does not work for vector valued
functions, so you need to make the line yourself. Here is a relatively
painless way to do this. Basically tanline is of the form slope*t + c,
where c is the point of tangency (here circ(t) evaluated at t=Pi/3). Note
that tanline is simply an expression, not a function. You should not type
tanline(t). You can arbitrarily choose your interval for the
plot of tanline to make it longer or shorter. Try Pi..Pi for
example.
> tanline:= evalm((scalarmul(slope,t)) +
circ(Pi/3));
> VPlot(tanline, t=0..Pi, color=BLUE, title=`tanline
of circ(t) at t=Pi/3`);
In order to display both plots on the same graph, you need to store the
instructions for plotting into a variable, then use the
plots[display] command.
> VPlotcirc:=VPlot(circ(t), t=0..2*Pi, color=RED,
axes=NORMAL,
labels=[x,y], title=`circ(t) with tangentline at t=Pi/3`):
> VPlottanline:=VPlot(tanline, t=2*Pi..2*Pi,
color=YELLOW):
> plots[display]([VPlotcirc,VPlottanline]);
For 3D graphs, you should use the right mouse button to rotate or pivot
the image for a clearer view. Use the middle mouse key to redisplay. To
compare two 3D graphs, you should rotate them all into the same position.
If, after rotating, you notice the scaling is different, you may want to
try Projection(Constrained) on the VPlot menu bar, in order to get an
accurate comparison. Also, choose a type of axes for reference e.g.
Normal.
Exercises
1. A particle's position vector at time t is
determined by
R(t) = 2i + 2j + 2k + (cos t) 

i  

j 

+ (sin t) 

i + 

j + 

k 

. 

(i) 

Plot and animate the graph of
R(t). 
(ii)  
Given the animated
picture, do you think this graph is substantially threedimensional or
does it all lie in a plane? 
(iii)  
If you feel the curve is planar, try to specify its plane. To this
end, consider an arbitrary point of the curve, say, the point
R(0) = 2i + 2j + 2k + 

i  

j, 

and attempt to find a constant vector n such that
[R(t)  R(0)]
· n = 0 for all
t. 

(iv)  
Find the curvature of the graph at the points t = 0, t =
/4, and t = /2. 
(v)  
On the basis of (iv), can you make a plausible hypothesis about the
genuine shape of the graph? 
(vi)  
If your answer to (v) is "yes", please specify your point and give a
detailed description of the curve. 
2. Explore graphically the behavior of the helix
R(t) = (cos at)i + (sin at)j +
btk

as you change the values of the constants a and b.
Set b=1. Plot the helix R(t) together with the tangent line to the curve
at t=3/2 for a = 1, 2, and 4 over the interval
0<=t<=4. Describe in
your own words what happens to the graph of the helix and the position of
the tangent line as a increases through these positive values.
Let a=1. Plot the helix R(t) together with the tangent line to the curve
at t=3/2 for b = 1/2, 1, and 2 over the interval
0<=t<=4. Describe
in your own words what happens to the graph of the helix and the position
of the tangent line as b increases through these positive values.
3. Explore graphically the behavior of the curve
R(phi) = (sin(phi)cos(n phi))i + (sin(phi)sin(n phi))j
+ (cos(phi))k,
0<=phi<=

as you change the values of n. Try values of 5, 10, and 15 for n
and animate the graph. It will also also help to use Projection
(Constrained) and Axis (Normal) to visualize the plot.
Given the animated picture, what surface do you think these graphs may
belong to? Justify your answer analytically.
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Created by Henry Fink
Last updated: Monday, September 29, 1997

