Cynthia Vaskis
SLM521 Spring
2004
Dropin #1 Assignment
How is math used to represent the real
world and objects in it?
Most math models of the physical world rely on a coordinate system to measure where things are in the real world. There are three main coordinate systems used to describe the world and objects in three dimensional space. The first one is the Cartesian coordinate system.
1. Go to this web site to learn who invented it and how he came up with the idea. Inventor of the Cartesian coordinate system.
a) Who invented the Cartesian coordinate system?
b) How did he first get the idea?
The 3D Cartesian coordinate system is used to describe objects in an Earth Centered Inertial (ECI) universe so that scientists can exactly locate objects on the surface of the earth and the position of objects orbiting the earth. The origin in the ECI system is at the center of the Earth and the positive X axis is in the equatorial plane pointing toward the first point of Aries or vernal equinox. The vernal equinox is the point on the first day of spring where the Sun passes through the equatorial plane heading northward. The equatorial plane is the plane that cuts through the equator all around the Earth and passes through the Earth’s center as well. See a picture of the Earth and the equatorial plane with the X axis pointing toward the vernal equinox in the middle of the explanation page of the Vernal Equinox and Equatorial plane.
2. The Cartesian coordinate system axes and
origin.
One of the axes is the positive X axis, one is the positive Y axis and one is the positive Z axis. Each of these axes also has a negative direction which must also be labeled. The X and Y and Z axes are all perpendicular to each other (called orthogonal) or at right angles to each other meaning there is a 90 angle between any two of them.
See the web site that describes the Cartesian coordinate system.
(NOTE: When the following example comes up move the right window edge to the left until the X axes (positive and negative) lines are one big line going through the origin (0, 0, 0) or where the Y and Z axes cross). Usually one draws the coordinate axes as in the Cartesian Coordinate System example.
The Right Hand Rule is used to determine the direction of the positive axes. The pointer finger on the right hand points outward horizontally (positive X axis), the middle finger points to the left horizontally (positive Y axis) and the thumb points upward (positive Z axis). In the next lesson (drop-in 2) we discuss rotating objects and we will use the Right Hand Rule to determine the positive angle rotation direction around each axis.
a) Practice drawing the Cartesian coordinate system on a piece of graph paper using the Cartesian Coordinate System example above, labeling the axes correctly and the origin.
b) What is the mathematical term (or word) used to describe the relationship or type of angle between the coordinate axes?
3. Locate points (or vectors) in the 3D coordinate system.
Each point in the coordinate system can be described by its X value, its Y value and its Z value. These values are found by measuring how far out the point lies in relation to each coordinate axis. A point (x, y, z) can also be described as a vector in the Cartesian coordinate system. It looks like an arrow that starts at the origin (0, 0, 0) and finishes at the point (x, y, z) in three dimensional space.
a) Locate the origin on your Cartesian coordinate graph from problem 2. a) and label it correctly as (0, 0, 0).
b) Pick 5 points in space and draw them on the same Cartesian graph used in problem 3. a) above as vectors (arrows) starting at the origin and ending at the point you have selected. Label the point with the (x, y, z) coordinates..
4. Define an object’s location in the coordinate
system.
a) Draw the X-Y-Z coordinate system axes on a new piece of graph paper using the previous Cartesian graph as an example.
b) Then draw a cube box (one unit long on each side) that sits on the z=0 plane with one bottom corner at the origin and the rest of the box extending into the positive x and positive y quadrants. Use the same graph paper from problem 4. a).
c) Give the (x, y, z) coordinates for each of the eight corners of the box.
( , , ), ( , , ), ( , , ), ( , , ),
( , , ), ( , , ), ( , , ), ( , , )
5) Translation of an object in the coordinate
system by adding or subtracting values along coordinate axes.
If I wanted to move the box around but keep its same orientation (the way its sides were facing) I would simply add the amount I wanted to move in that direction to the coordinate that defines the value in that direction. See HintP5 for the corner point vectors.
a) Move the box in problem 4. c) side-ways in the negative X direction by one unit. Give the new coordinates below. (Hint: Add a minus one (-1) to each of the X values of the box’s corner points (x, y, z)). Use the beginning position of the box as defined by the coordinates from problem 4. c).
( , , ),
( , , ), ( , , ), ( , , ), ( , , )
b) Using the result from problem 5. a), move (or translate) the box another 2 units in the positive X direction, 3 units in the positive Y direction, and 1 unit in the positive Z direction. Give the new coordinates below. See the HintP5b page for the solution. This is equivalent to adding the vector (2, 3, 1) to each point vector of the box’s corners. Adding two vectors is done by adding their “x” values to get a new “x” value, then adding their “y” values to get a new “y” value and adding their “z” values to get a new “z” value. (x1, y1, z1) + (x2, y2, z2) = ((x1+x2), (y1+y2), (z1+z2)) or is the new vector (x3, y3, z3) where x3 = x1 + x2, y3 = y1 + y2, and z3 = z1 + z2. ?
( , , ), ( , , ), ( , , ), ( , , ),
( , , ), ( , , ), ( , , ), ( , , )
c) If I wanted to move the box only in the positive “y” direction by 3 units then I would add the number 3 to each of the “y” values in the box’s point vectors which would be adding the vector (0,3,0) to each of the box’s corner vectors. Notice the 3 is in the “y” position in the vector and there are zeros in the other positions if we are only moving in the “y” direction and are not moving in any other direction. What would be the coordinates for each corner of the box from problem 4. c) if I picked the box straight up by 2 units (which coordinate axis is in the up direction) and moved it in the negative X direction by two units and moved it in the positive Y direction by 3 units? Put the point vector values for the box’s new location below. HintP5c
( , , ), ( , , ), ( , , ), ( , , ),
( , , ), ( , , ), ( , , ), ( , , )
All of the answers to these problems can be seen on the Answer Sheet.
Many times data used in space applications is generated in different coordinate systems and that data needs to be converted to a common system in order to perform rotational maneuvers of the objects in orbit or space. The Kepler equations define an object’s orbital parameters or orbital path at any point in time. An object or person on the Earth’s surface may be defined in Polar coordinates as the Earth’s surface locations are usually in latitude and longitude coordinates. Often, data from these two different systems is converted into the ECI coordinate system mentioned above. The ECI system is a special case of the Cartesian coordinate system because its origin is the center of the Earth and the coordinate axes do not rotate with the Earth but remain fixed in 3D space. We will not cover these other two topics (Kepler equations and Polar coordinates) until a later lesson. It might be a good idea to see what the other two major coordinate systems look like which are the Polar coordinate System and the Spherical coordinate system (mouse click to read about them). For our purposes we will mainly work with an ECI Cartesian coordinate system since many of the real world examples use that system.
SUGGESTION: This lesson can be demonstrated to students using a 3D model of the coordinate system axes made of three pencils or straws taped together in perpendicular angles.
We have covered defining an object’s position in the Cartesian coordinate system and moving, or translating, it to some other place in the system without changing its orientation (the direction the sides of the box are facing). This type of math will be used to figure out where an object is located in space (its position vector on its orbital path) and where smaller objects are located on the bigger object (translation calculations). This is nice but to really be useful we must understand how to rotate the object (or box) so that it has a new orientation (its sides face a different direction). We will cover that in the next lesson (dropin2).
Some Math HELP web sites are listed below:
Mathworld – Probably the most complete reference guide to mathematical definitions on the web today. Every math definition possible is listed here with some problems and solutions. It is geared more toward a professional engineer or mathematician but if a student wanted a clear definition of a topic in math, it could be found here.
http://mathworld.wolfram.com/about.html
The Math Forum – Ask Dr. Math – Tutorial on elementary, middle, high school, and college level mathematics which includes problem solving, formulas, FAQs, archive search, and permits questions via email to Dr. Math. This site has math problems and help for all elementary and secondary math levels. It would be a good site to review topics to study for a math test.
http://mathforum.com/dr.math/