Lab #2
This worksheet does not need to be handed in, but you are responsible for the material. The ideas laid out here will be used in next week's lab (which will be graded).
The purpose of the lab is
Log in to your account and launch Maple. The first command
you should type into Maple is
with(linalg): with(plots):
In this lab, we will study planes. Planes are the simplest surfaces, and as such they are often used to locally approximate more complicated surfaces. The plane that best approximates a surface at a point is called the tangent plane at that point, in much the same way that a tangent line is a local approximation to a curve.
As is the case with many surfaces, there are three ways to represent planes: as a graph, as a parametrized surfaces, and as an implicitly-defined surface. In this lab we will concentrate on parametrizing planes, because this will lead most naturally (next week) into our study of tangent planes.
Define the following vectors in Maple:
T1 := vector( [1,0,0] ); T2 := vector( [0,1,1] );(Note that Maple's output does not include commas between components of its vectors; this can sometimes be confusing.)
We want to form the span of these two vectors. Algebraically,
the span of two vectors is the set of all vectors that are linear
combinations of the two vectors. In symbols, it is the set of all
vectors that may be expressed as 
where s and t are
allowed to take on any real value. Perhaps easier to visualize is
the geometric interpretation: the span of two vectors is the set of
points you can reach by traveling some distance along the first
vector, then turning in the direction determined by the second vector,
and traveling some distance in that direction.
Let's parametrize a plane in Maple. Define the plane as the
image of the map 
by using the add
command (you may want to look at ?add):
Plane := add( T1, T2, s, t );Make sure you can explain your answer! The variable Plane is now an expression in the parameters (s,t), so we can plot (a portion of!) the plane by
plot3d( Plane, s=0..1, t=0..1, axes=boxed, labels=[x,y,z], view=-4..4);(Note that the last two options are not necessary, but help you to visualize the 3D picture easier.)
and 
?
(Hint: look very carefully at the tick marks on the axes, and compare
with 
and 
.)NewPlane := add( [0,0,1], Plane ); plot3d( NewPlane, s=0..1, t=0..1, axes=boxed, labels=[x,y,z], view=-4..4);What was the effect of adding the vector to the original plane? (Again, look very carefully at the tick marks on the axes!) Try adding the vectors (-1,-1,-1) and (-2,-2,0) to Plane. Can you explain how adding a constant vector changes the points that the plane passes through?
depends on the values of the vector p
and the parameters s and t. In this question, we ask you to vary the
direction vector 
to get a feel for how the plane depends on it.
Redefine
T2 := vector( [1,0,1] ); Plane := add( T1, T2, s, t );and plot the plane as before with
and 
.
Can you determine how the plane changed? Conjecture how the plane depends on .
Test your conjecture by reentering the above commands so that
and then 
. What is happening to
your plane as get closer and closer to 
? Conjecture what
would happen if 
. Test your conjecture. (Hint: For easiest
comparison, keep all four plots of this question on the screen at the
same time. Turn the images so that the x-axis point to the right of
the screen, the z-axis point to the top of the screen, and the y
axis point ``into'' the screen.)
The geometric meaning of two vectors being linearly independent is that the span of the two vectors is a plane. If their span is only a one-dimensional line, we say the two vectors are linearly dependent, or that one vector is a multiple of the other.