# Euclidean Space

The real numbers are denoted by . I'll assume you're familiar with the basic properties of , but I'll mention less familiar things as they come up.

The standard 2-dimensional plane is denoted . It consists of ordered pairs of real numbers:

You're probably familiar with how a point is located in the plane:

The "x" and "y" on the axes label the positive x-axis and positive y-axis.

Similarly, 3-dimensional space is denoted . It consists of ordered triples of real numbers:

(There are also , , and so on, defined in similar fashion.)

Just as it's conventional to use x and y to denote the first and second coordinate variables in , it is conventional to use x, y, and z to denote the first, second, and third coordinate variables in . The picture shows how a typical point is located in space:

The "x", "y", and "z" on the axes label the positive x-axis, the positive y-axis, and the positive z-axis.

At this point, I should note a convention that we'll always follow. The labelling of the positive axes in and follow the "right-hand rule". In the x-y-plane, you curl the fingers of your right hand from the positive x-axis to the positive y-axis through a angle. Check for yourself with the picture above.

(Note that in some computer graphics applications --- for instance, in the SVG graphics language --- it's conventional to have the positive y-axis going "downward" rather than "upward".)

For , curl the fingers of your right hand from the positive x-axis to the positive y-axis. As you do so, your thumb points in the direction of the positive z-axis. Check for yourself with the picture above.

Graphing things in 3 dimensions is obviously harder than graphing things in 2 dimensions. While you don't need to have great artistic skills, you should practice making diagrams in 3 dimensions as they are a huge aid to understanding. Short of taking a drawing class, the best approach might be to copy other peoples' pictures until you get the idea. For starters, you can practice plotting points in .

Example. Plot the points , , and . The pictures below aren't perfectly scaled; I just want to locate the points in approximately the right places.

The axes and the coordinate planes. Consider the x-axis. It is perpendicular to the y-z plane, and passes through the origin of the y-z-plane. Therefore, it is determined by the equations

I'll discuss lines later on, and in particular their equations in parametric form. The parametric equations for the x-axis are

Similarly, the y-axis is

The z-axis is

Now consider the x-y plane. It consists of all the points at "z-level" zero --- that is .

Likewise, the y-z plane is , and the x-z plane is .

Distance. The distance between points and is given by

(You can write " ", and so on, instead; the squaring makes the order of subtraction irrelevant.)

Here's where the formula comes from.

The box has sides of lengths , , and . By Pythagoras' Theorem, the diagonal s of the bottom of the box is

I can drop the absolute values because I'm squaring the terms.

Again by Pythagoras' Theorem, the distance is

Example. Find the distance between the points and .

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