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Cramer's Rule
A slightly different explanation from our text, click here to view. From the
Educational Tools Company. For a biography of Gabriel Cramer, click here.
The biographical link is to the web site of Trinity College at Dublin, Ireland.
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Systems of Linear Inequalities
The Department of Mathematics at Michigan State University
in East Lansing, Michigan has posted math class lectures on their web site
(Lectures 10 - 22 are online).
You can watch and listen to a lecture, with graphs, of a
presentation by Dr. Mary Jean Winter of MSU. This one is approximately 17 minutes long. You may need to
download the Real Player Plugin for your browser. Click here to
go to the Systems of Linear Inequalities (Lecture 19) page. Below are more lecture links (all the ones I could find). Not all
of these are relevant to our course, but a lot of them are. You can see the material presented from a different viewpoint,
all you have to do is fix some popcorn, sit back and watch:
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Graphing Equations in Three Variables
This Manipula Math applet allows you to play with a "live" X, Y, Z set of axes.
Using your mouse, you can drag the axes to whatever position you want so you
can see your graph from any viewpoint - it it very well done. The X, Y, and Z axes
are not labeled.
This tool can help you to visualize the exercise (3-7B on pages 172-173) we did in class.
Note that the box for entering the equation you want to graph is of the form:
Z=_______. So you need to put your equation in this form before you can use the
live graphing feature. For example, Activity 2 is: Graph 3X + 4Y +3Z = 12. Solve this
for Z which yields: Z = (-2/3)X -(4/3)Y +4. Enter everything on the right-hand side of
the equation into the box. Try this one to see if you get the
same plane you graphed by hand. We only plotted the intercepts so our graph looked like
a triangle, the plane you see drawn by Manipula Math has more points and it looks like
a rectange (our triangle is on this larger plane's surface). Remember, the plane extends to infinity
in all directions. You may want to graph and play with some simple planes first,
for example Z=1. Don't forget to try rotating the axes. Click here to go to the Manipula Math program, it is
called "Simple 3d Graph Z=f(x,y). Many functions we have not covered are listed,
plan to revisit this site as we learn about these functions.
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3-D Drawing and Geometry by Cathi Sanders
An explanation of general 3-D (3 dimensional) drawing that includes an explanation
of Isometric Drawing. Check out the isometric drawing examples. The link
here starts you out at the Isometric Drawing chapter, the rest of the tutorial is
quite interesting as well. Click here to go to the Isometric Drawing chapter.
Keep in mind the goal, to use a flat surface (your paper) to show a three dimensional object.
Some day your TI-186 holographic calculator will show your graphs in 3-D floating over your
desk, until then we need some way to display our X, Y, and Z axes on a flat surface.
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Isometric Drawing -- Definition
Source: Encyclopedia Britannica Online: http://www.britannica.com/
Also called Isometric Projection, method of graphic representation of three-dimensional objects,
used by engineers, technical illustrators, and, occasionally, architects. The technique is intended
to combine the illusion of depth, as in a perspective rendering, with the undistorted presentation
of the object's principal dimensions, that is, those parallel to a chosen set of three mutually
perpendicular coordinate axes.
The isometric is one class of orthographic projections. (In making an orthographic projection,
any point in the object is mapped onto the drawing by dropping a perpendicular from that point
to the plane of the drawing.) An isometric projection results if the plane is oriented so that it
makes equal angles (hence "isometric," or "equal measure") with the three principal planes of
the object. Thus, in an isometric drawing of a cube, the three visible faces appear as equilateral
parallelograms; that is, while all of the parallel edges of the cube are projected as parallel lines,
the horizontal edges are drawn at an angle (usually 30º) from the normal horizontal axes, and
the vertical edges, which are parallel to the principal axes, appear in their true proportions.
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