Points, Lines, Angles, and Planes
Remember back when we were musing philosophically about dimensions and slices and all that jazz? We weren't kidding about the philosophy. Much of that stuff was developed by Rene Descartes (famous for the oh-so-overquoted, "I think, therefore I am"). And he wasn't even Greek!
Descartes realized that coordinates can specify points, and geometrical relationships between the points can be interpreted as numerical relationships between their coordinates. That means we can have two coordinates and find the distance between 'em.
Let's start with a simple example: say we have a horizontal line l containing a point O.
We can give O the coordinate 0 (we sometimes call this starting point the origin). If point P is 5 units to the right of O, we can give P the coordinate 5. In general, we can use positive numbers for right or up, if the line is vertical, and negative numbers for left or down. Then, we just write down how far the point is from O. If the line is diagonal, pick positive to be the direction that tends towards the right.
What is the coordinate for point X in the picture?
Since X is 3 units away from O, and is down from it, we should give X the coordinate -3.
Putting coordinates on lines really just makes them look like the familiar number line from algebra.
Then, finding the distance between two points on the line is easy: it's the bigger coordinate minus the smaller coordinate. Distance should always be positive, so we tend to stick the answer in an absolute value sign just in case we subtracted the wrong way.
Suppose X has coordinate 4, and Y has coordinate -1. What is the length of the segment XY?
Let's start by drawing a picture:
Since the length of XY is just the distance from X to Y, we can compute it from the coordinates as
XY = |4 – (-1)| = |5| = 5.
What if we're trying to pinpoint a location on a piece of paper using coordinates? We can pick, say, the upper left corner as the origin, calling it O. But which point should get the coordinate 1? There are too many points of distance 1 from the origin to pick from.
We can't even get around this by using +1 and -1, since there are more than two options. However, we can pinpoint exactly one location by using two coordinates. For example, we can say point P is 0.8 units to the right and 0.6 units down from O.
This specifies P exactly, and we can say (0.8, -0,6) are the coordinates of P.