### Information for Maple assignment 1

Each student will get individual data for this assignment.

Here is some help for the first Maple assignment. The assignment requests several pictures. Pictures are very important. Not many people can get much information from vast tables of numbers, but humans possess a large capacity to receive and organize visual information. You are allowed, and indeed encouraged, to copy and modify the commands discussed here. Remember that all commands should be entered into Maple in one line - this page has some commands broken across two lines because of space limitiations.

### Packages

Maple has thousands of commands, but many of them are kept in storage in "packages" to save memory. Once Maple is started, you can call up packages via the with command. One package, called plots, contains several three dimensional plotting commands. Enter with(plots); in Maple to see a large number of commands appear which are now easily accessible. One of them is spacecurve which will plot three-dimensional parametric curves. To see more about this command you can type help(spacecurve); to bring up the help page.

### Beginning pictures

The command
spacecurve([4*t-11,3*t+7,-5*t+2], t=0..1, axes=normal, color=black, thickness=2, labels=[x,y,z]);
draws the parametric function
x = 4t - 11
y = 3t + 7
z = -5t + 2
for the range 0 ≤ t ≤ 1. These equations should give you a line segment from the point (-11,7,2) where t=0, to the point (-7,10,-3) where t=1.

The picture Maple displays (shown above) is deceptive. The line segment is not vertical although it is shown that way. In this case this problem arises because the three coordinate axes are not drawn to the same scale.

Add the "scaling=constrained" option to change the command to
spacecurve([4*t-11,3*t+7,-5*t+2], t=0..1, axes=normal, color=black, thickness=2, labels=[x,y,z], scaling=constrained);
The result is shown below. The scaling=constrained makes all the axes have the same scale, and so the line segment is no longer displayed vertically.

The command
spacecurve([3*cos(t)+10,3*sin(t)+4,-3], t=0..Pi, axes=normal, color=black, thickness=2, labels=[x,y,z]);
draws the parametric function
x = 3 cos(t) + 10
y = 3 sin(t) + 4
z = -4
for the range 0 ≤ t ≤ π. (Notice that to get the constant π you need to use the capitalization Pi. Maple treats the words pi and PI as variables.) This should be a semicircle of radius 3 centered at (-10,-4,-3) in the plane z=-3. The view below is unconstrained. The result seems to be a parabolic arc because Maple attempts to stretch the curve to fill up the viewing window as much as possible.

Change the command to
spacecurve([3*cos(t)+10,3*sin(t)+4,-3], t=0..Pi, axes=normal, color=black, thickness=2, labels=[x,y,z], scaling=constrained);
The result is shown below. The curve looks much more like a semicircle.

The image can be manipulated in various ways before "exporting" or printing it. It can be rotated, scaled, etc. with some mouse clicks on the picture.

### Displaying Graphics Together

Often one will want to draw multiple graphs on the same axes for easier comparison. In Maple this is achieved via the display command found in the plots package. Store each graph to its own variable, and then display the set of those variables. For example,
A:=spacecurve([4*t-11,3*t+7,-5*t+2], t=0..1, axes=normal, color=black, thickness=2, labels=[x,y,z], scaling=constrained):
B:=spacecurve([3*cos(t)+10,3*sin(t)+4,-3],t=0..Pi, axes=normal, color=black, thickness=2, labels=[x,y,z], scaling=constrained):
display({A, B});

The first two commands assign values to the variables A and B. The plots are what get stored to the variables. The display command shows the plots together. As is true with plotting single curves, you'll need to be careful in creating accurate unambiguous graphs. The next section describes some problems to avoid.

### The Importance of Good Views

We will now show several views of the same pair of curves given above. We show very bad versions of the pictures to emphasize that poor graphs can actually decrease the effectiveness of technical communication instead of helping.

This is the plot with scaling=unconstrained.

This is the result with scaling=constrained.

Here is a a very bad version of the picture. It is taken from "the side", with the x-axis coming straight out of the image. This picture seems to show two line segments, when one of the segments is actually the semicircle viewed edge-on.

Here's another very bad version of the picture. The axes have been taken away, and the viewing angle makes the line segment seem to cross what could be a parabolic arc.

### Analyzing Vectors

Suppose p is the point (3,10,-7), q is the point (9,8,3), and r is the point (6,5,7). Then we can get the vector from p to q via:

```[9,8,3] - [3,10,-7];
[6, -2, 10]```

This is the vector from p to q. Notice that square brackets [] are used to represent points/vectors instead of parentheses. A picture of the vector can be created with the spacecurve command. We will store that plot to the variable PQplot.
PQplot:=spacecurve([6*t+3,-2*t+10,10*t-7], t=0..1, axes=normal, color=black, thickness=2, labels=[x,y,z], scaling=constrained);

A similar computation gets the vector from p to r and the corresponding picture:

```[6,5,7] - [3,10,-7];
[3, -5, 14]```

PRplot:=spacecurve([3*t+3,-5*t+10,14*t-7], t=0..1, axes=normal, color=black, thickness=2, labels=[x,y,z], scaling=constrained);

What about the cross product of the two vectors? The VectorCalculus package has a CrossProduct command with a short version &x. When using the VectorCalculus package, denote vectors via angle brackets < and >.

```
with(VectorCalculus):
<6,-2,10> &x <3,-5,14 > ;
22 ex  - 54 ey  - 24 ez
```
What are these ex, ey, ez? They are the elementary vectors in the x, y, and z directions, respectively. These are exactly the same as what your book calls i, j, and k. Thus 22 ex -54 ey -24 ez means the vector < 22, -54, -24 >.

We then can draw the cross product vector as a line segment "based" at p:
CPplot:=spacecurve([22*t+3,-54*t+10,-24*t-7], t=0..1, axes=normal, color=black, thickness=2, labels=[x,y,z], scaling=constrained);
and all three graphs can be displayed with
display3d({PQplot,PRplot,CPplot});

The above picture was rotated so that the cross product vector is more clearly perpendicular to the other two. Other views, such as the one below, are not nearly as clear and should be avoided.

Here is an even worse view: one vector is entirely hidden behind one of the others, making it look like there are only two vectors drawn.

Below is a picture of the triangle in R3 whose vertices are the points p and q and r. This picture is very easy to create using a command in the plots package. Find this command yourself: look at the list of commands in plots, guess, and then use help. A hint: The command you're looking for can do more than triangles. It can also plot quadrilaterals, pentagons, hexagons, and many other two-dimensional shapes in three dimensions.

By the way, by rotating the viewpoint, you can make any of the angles of this triangle seem to be a right angle! Perspective can be misleading and irritating.

Finally, here are the triangle, two of the vectors along the triangle's edges, and the cross product of these two edge vectors. You would also generate this picture via the display command. The view has been chosen so that the cross product appears to be perpendicular to the triangle, which it is. Twice the area of the triangle is equal to the length of the cross product.