Coordinate Systems

From High School Online Collaborative Writing

Contents 1 Coordinate Systems Overview 2 Coordinate Systems With Vectors 2.1 Example Problem 3 Different Types of Coordinate Systems 3.1 Spherical Polar Coordinates 3.2 Coordinate Systems in 3-D 4 Resources 5 References

[edit] Coordinate Systems Overview

A coordinate system is a system that measures distance and direction. The coordinate system consists of different coordinates and an origin (usually the reference point). The coordinates refer to the direction of the motion while the origin refers to the magnitude of motion. The coordinate system shows where the starting point of the variable is, and which directions the variables will increase. The origin is where the variables have a value of zero. For an example, look at Fig. 1 which shows a diagram of two runners. If you were to try and measure the distance and time of these two runners, the first thing you'd have to decide is where to start measuring. The origin is like the starting line, so to start measuring the distance; a measuring tape should be placed at the "starting line". The runners are moving towards the left in a straight motion, therefore the measuring tape should put on a straight line, which is the X axis of the coordinate system. In this scenario the Y axis has no use, but in some cases the vertical distance is an important factor.

[edit] Coordinate Systems With Vectors

Coordinate Systems not only act as a system for measurements, but they come in very handy when adding Vectors. But before adding the vectors, you have to choose a coordinate system. The direction of the axes depends on where the measured quantity physically is. If the motion takes place on the surface of the Earth, the Y-axis points North, and the X-axis points East. Like the surface of the Earth, if the motion is through the air the Y-axis points vertically and the X-axis points horizontally. If the motion is on a hill, the X-axis is placed in the direction of the movement, and the Y-axis is always placed perpendicular to the X-axis. When placing a vector on the graph, the direction should correspond with the direction of the axes.

In this coordinate system (Fig.2), the vector A is broken up into two more vectors. As you can see, vector Ax + vector Ay will lead you too vector A. The magnitude and signs of these vectors are called the components. Whether the component is positive or negative all depends on which quadrant the component is in. In each quadrant X and Y have different values. In quadrant I, X is greater than 0 and Y is greater than 0. In quadrant II, X is less that 0 but Y is greater than 0. In quadrant III, both X and Y and less than 0 and in quadrant IV, X is greater than 0 but Y is less than 0.

[edit] Example Problem

The question is: A bus travels 23.0 km on a straight road that is 30 north of east. What are the east and north components of its displacement?

Now normally this question would be solved algebraically, but in terms of coordinate systems, it's important to know what direction these vectors are going in and whether they are positive or not. In the problem we see that the bus travels north of east. East is in this direction > and north is in this direction ^. By this we know that the vectors will be in the first quadrant making them positive.

[edit] Different Types of Coordinate Systems

[edit] Spherical Polar Coordinates

The normal coordinate systems that we are used to seeing are usually of linear relationships. Vectors are definitely going to be straight lines, and very safe with their magnitude and direction. But not all coordinate systems use linear functions. An example for using the spherical coordinates would be when talking about longitude or latitude, because the Earth and many planets are spheres.

[edit] Coordinate Systems in 3-D

Coordinate systems are not only reserved for the second dimension, but appear in the third dimension as well. The 3-D coordinate system has 3 physical dimensions; length, width and height. Like the the 2-D graphs, the 3-D graph had the coordinates X and Y. But this time a third variable is added. Also similar to the 2-D system, the planes divide the space into eight sections called octants.

[edit] Resources

For more help or practice with coordinate system related questions, check out these following websites:

1. Regents Prep Answer some multiple choice questions.
2. Vectors and Coordinate Systems A quick little quiz with questions about vectors and coordinate systems.
3. Coordinate Systems Feeling adventurous? Read about coordinate systems that aren't on paper.

[edit] References

1. Cartesian Coordinate Systems
2. Coordinate Systems in Physics
3. Lazar, Miriam A. Physics: The Physical Setting 3rd. North Babylon, NY: Barron's Educational Series, 2004.
4. Zitzewitz, Paul W. Physics Principles and Problems Westerville Ohio: Glencoe, 1999.
5. Spherical Coordinates