3D Coordinates
Key Questions

We live in a 3D world, so 3D Cartesian coordinates are used to describe problems in our world. However, until we have holographic displays, the way we normally display information is 2D, such as paper or monitors. So 3D is normally projected onto 2D; this could be a perspective projection or orthographic projection.
The most common uses of 3D coordinates are in engineering, physics, and computer graphics. This would be done with software such as AutoCAD, Maya, SoftImage, 3DS Max, and Blender. As you know, computer animated movies can be shown in 3D or 2D, but both still have perspective. In order to have perspective projections, it is required to model everything in 3D.
Motion capture is done in 3D, so that they can be put in computer games or special effects in movies as well as the animated movies.
In engineering, buildings or bridges or any structure, require 3D so that workers know how wide, long, and high to build things. Another use that you may have heard about is 3D printers; obviously you need 3D coordinates to send to the printer for it to build anything meaningful.
There are some other uses that you may know such as topographical maps that show the height of a 2D position or an air traffic display will show the 2D position of an aircraft, but display its altitude beside the position.
These are just a few of many examples.

If Cartesian coordinates are
#(x,y,z)# , then its corresponding cylindrical coordinates#(r,theta,z)# can be found by#r=sqrt{x^2+y^2}# #theta={(tan^{1}(y/x)" if "x>0),(pi/2" if "x=0 " and " y>0),(pi/2" if " x=0" and "y<0),(tan^{1}(y/x)+pi" if "x<0):}# #z=z# Note: It is probably much easier to find
#theta# by find the angle between the positive#x# axis and the vector#(x,y)# graphically.
I hope that this was helpful.

For much of math, we speak about a 2D Catersian coordinate system with just the (x,y) components. In that system, you can only go horizontally in the xdirection a certain amount and vertically a certain amount in the ydirection. With the 3D Caterisian coordinate system, it's taken one step further There is a new direction, the "z" direction, which indicates a more realistic coordinate system. The three dimensions of x,y, and z indicate positions of lines/points that can actually be modeled in the real world.
To better visualize this, you can think about a page representing the 2D coordinate system (ignore the thickness of the page so that it seems 2D). On that page, you can only draw lines that lie directly on that page. They have only the dimensions of x and y. However, with 3 dimensions, the "z"component can indicate whether that point is situated "into" or "out of" the page. Imagine poking a thin stick through the page. That could indicate your z axis, which adds a whole new dimension to the illustration.
This is better explained with a picture. This is a picture of a typical Cartersian Coordinate system:
Note that the orientation of the xyand z axes can change according to convenience of use, but these are just to illustrate how the 3D factor of the Cartesian system comes into play. There are three axes to define position with a 3D coordinate system, unlike the singular xy plane in the 2D coordinate system.