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

1. to use technology to visualize planes in space;
2. to understand how planes are defined by a point and two linearly independent vectors;
3. to understand the geometric significance of each algebraic term in a parametrization of a plane.

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.)

1. Vary the range of s and t in the plot3d command. What happens if s and/or t are allowed to take on negative values? What portion of the plane do you get if and ? (Hint: look very carefully at the tick marks on the axes, and compare with and .)
2. Currently, the plane is ``centered'' at the origin (0,0,0) in the sense that the plane passes through the origin when s=t=0. Now we will translate the plane by adding some vector to the plane:
```	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?
3. At this point, you should understand how the image of the map 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.