3D Rectangular Coordinates

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3-Dimensional Coordinates

Based on “Three Dimensional Coordinate Systems,” “Distance in R³,” and “Writing Equations in R³” in section 1.2 of the textbook.

Key Ideas

3-dimensional coordinate systems are defined by 3 perpendicular axes. Each possible pair (xy, xz, yz) defines a “coordinate plane” (which you can think of as having a 2D coordinate system in it, for points in which 2 of the 3D coordinates can vary but the third is always 0). The coordinate planes split space into 8 “octants,” analogous to the 4 quadrants in 2D coordinates.

There’s a distance formula for 3D space, which really just adds a z term to the formula for 2 dimensions:

This distance formula is really an application of the Pythagorean Theorem, although it might have a different name because the Pythagorean Theorem is specifically about triangles in 2 dimensions, while the distance formula applies to the distance between points in any number of dimensions. And as someone mentioned to me after class, the connection between triangles and distances between points may not be obvious if no-one has pointed it out to you before: the “x₂-x₁,” “y₂-y₁,” etc. terms in the distance formula are the lengths of sides of triangles, for example like so in 2 dimensions:

Many things you’re used to from 2 dimensions scale up in natural ways to 3 dimensions. The distance formula is a good example. Another is the equation for a sphere, which is just like the equation for a circle, but with a “z” term added. But there are also sometimes new behaviors that the third dimension makes possible, for example two non-parallel lines that don’t intersect.

A third dimension also lets you create shapes that are infinite in some dimensions but not others, in a richer set of ways than you can in 2 dimensions. For example cylinders that are closed circles in 2 of the dimensions but infinitely long in the third.

Questions

How to read and create 3 dimensional graphs?

Creating them is best done with a calculator on computer tool. I’ll demonstrate one, Mathematica, on Monday.

To help with understanding what they mean, we tried locating some points in the classroom relative to a physical 3D coordinate system.

• The center of the top of an occupied table was at about (-2, -2, -0.5), i.e., 2 units left of the origin (the positive x axis pointed right), 2 units in front of the origin (the positive y axis pointed away from the class, forming a “right-handed” coordinate system), and half a unit below the origin (the positive z axis pointed up).
• We estimated that one corner of the ceiling was at about (10, 4, 5).

(There’s no particular reason why the y axis has to be horizontal and the z axis vertical, that’s just the most common convention. Some math communities, notably computer graphics, do usually think of y as up and z as forward/backward.)

We also thought about some simple equations in 3 dimensions. For example the equations x = y = z define a straight line through the origin that makes 45 degree angles with each of the 3 axes. A sphere with radius r and centered at (a, b, c) would have equation (x-a)² + (y-b)² + (z-c)² = r².

Examples

Find the distance between the table top and ceiling corner points mentioned above. Plugging the points into the distance formula (it doesn’t matter which you consider the starting point and which the ending) gives…

Problem Set

The first problem set is available. It covers 3D coordinate systems, and working with Mathematica (which I’ll introduce on Monday).

Work on it through next week, and grade it the week after.

See the handout for more information.

Next

An introduction to Mathematica, with particular attention to plotting in 3 dimensions with it.

There’s no reading, but…

Please be sure that Mathematica is installed on your computer by Monday. Instructions for doing it are here, and there’s a link to them from the syllabus.

Please bring the computer to class so you can try out things I show.

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