Line and ray: what changes when you refuse to stop

0:00Last time we built a line segment. Two points, A and B, and the shortest route between them. It has ends. It stops. That was the last thing I said. So now let's do the obvious thing, and refuse to let it stop. Push the stroke past B. Keep going. Further. Don't stop. Now push it past A as well, the other way. Keep going there too. What you've got now is called a line.

0:28And notice what it is. It's the segment, with the stopping taken away. That's the only change. But that one change buys you something quite big, and we'll get to it in a minute. First, a question your book actually asks, and I like that it does. Could you draw the whole line, if you had a big enough sheet of paper? Think about it properly before answering. The answer is no. And not because paper is expensive.

1:00It's no because the endlessness is in the definition, not in the paper. A line doesn't run a very long way. It runs without end. There is no sheet big enough, because there's no length to fit. So every drawing of a line you will ever see is incomplete. Yours, mine, your textbook's. Which is exactly what the arrowheads are for. An arrowhead isn't decoration, and it doesn't mean straight.

1:29It means: this continues, and I have simply stopped drawing. That's also why a segment drawn straight has no arrowheads. It really does stop. Now, how do you write these down? Your book gives you a set of marks, and they're worth learning properly, because each one records its definition. A segment through A and B: the two capitals, with a plain bar over the top. Flat at both ends, like the object.

2:01A line through A and B: the same two capitals, with a bar that has an arrowhead at each end. Open at both ends, like the object. So the notation is a tiny picture of the thing. There's also a shortcut for lines. You can give the whole line a single small letter. Little l, or little m. Your book draws a line through A and B and writes m next to it, so you can see both names on one picture.

2:31That's handy when you don't care which points are on it — you just want to call it something. Right. Here's what taking both ends off actually bought us. Mark two points. Now draw a line through both of them. Try to draw a second, different line through both of them. Go on, really try. Tilt it a bit. You can't. Once you've fixed two points, there's exactly one line through them both. One. Not two, not several.

3:02Your book uses the word unique for this, and unique means exactly one. And it's worth seeing how strong that is. Two dots on a page determine an infinitely long object, completely. Everything about that line — every direction it goes in, every point it passes through, for ever — is decided by two dots. That's a claim a segment could never make, because there were infinitely many routes between the same two points.

3:36Your book puts two students side by side here, and the contrast is the whole point. Rihan marks one point on a sheet, and asks: how many lines can pass through this point? Sheetal marks two points, and asks the same question. Take them one at a time. Sheetal's first, because we just did it. Two points. Exactly one line. Now Rihan's. One point. Draw a line through it. Now another one, at a different angle. Through the same point.

4:09And another. And another. You can spin a line round that point for ever and it still goes through it. So through one point: unlimited. Endlessly many. Through two points: exactly one. That's a strange pair of answers if you don't look at why. So here's why. The first point says the line has to go through here. That still leaves the direction free, and there are endless directions. The second point takes the direction away too. And once position and direction are both fixed, there's nothing left to choose.

4:48Now the other move. Instead of taking both ends off, take just one. Start with a line. Pick a point on it, and cut there. Throw one half away. Keep the other half, arrowhead and all. What's left starts at that point, and runs on for ever in one direction. That's a ray. The point where it starts has a name. Your book calls it the starting point, and also offers initial point. Both are used.

5:18And the notation follows the object again. Two capitals, with a bar that has one arrowhead. Closed at one end, open at the other. Rays are the one thing in this chapter you've actually seen. A lighthouse. The beam starts at the lamp, and heads out to sea. A torch. The beam starts at the bulb, and goes wherever you point it. Sunlight. It starts at the sun, and arrives here.

5:48In every one of those, there's a definite place where it begins, and no definite place where it ends. That's the shape of a ray. A start, and then an open direction. Now, being honest with you — a real torch beam does stop eventually, and it spreads out, and it's not perfectly straight. It's a model, like the pencil dot was. Good enough to think with, not the thing itself.

6:17Now naming, and this is where people lose marks, so let's be careful. A ray is written with two capitals, and the starting point goes first. Always first. So a ray that begins at A and runs out through P is called ray A P. The first letter tells you where it starts. The second letter just tells you which way it goes. Let's test that on a figure from your book.

6:44There's a point T, with two paths leaving it. One runs up to the right through a point A. The other runs to the right through N, and then on to B. So which rays are there? Ray T A — starts at T, goes through A. Ray T N — starts at T, goes through N. Ray T B — starts at T, and B is further along that same path.

7:08And one more, and this is the one people miss. Ray N B. That one starts at N, not at T. Three of the four start at T. The fourth doesn't, and you can only tell by reading the figure, not by reading the alphabet. Here's something that follows straight from that, and it surprises people. One ray can have lots of different names. Your book draws a ray leaving a point O, and it passes through B first, and then A.

7:41You can call it ray O A. Or you can call it ray O B. Both are correct. Both name the same ray. Because the second letter isn't picking out the end of anything. The ray has no end. The second letter is just saying: it goes this way. And any point lying further along the ray does that job perfectly well. So one ray, many names. All of them starting with the same letter.

8:10But there is one swap you are not allowed to make, and this is the trap the last video set up for you. Last time we said segment A B and segment B A are the same segment. Same points, either spelling. So you'd expect ray O A and ray A O to be the same too. They are not. Ray O A starts at O and runs out through A, and keeps going that way.

8:37Ray A O starts at A and runs out through O, and keeps going the other way. Different starting points. Opposite directions. Two completely different rays that happen to lie along the same line. So here's the rule to remember. For a segment, the order of the letters doesn't matter. For a ray, it matters completely. And the reason isn't a convention somebody chose. It's that a ray has a direction, and a segment doesn't.

9:09Which was the whole point of taking one end off instead of two. Let's finish by reading a figure that has all of it at once. Here's a straight path with arrowheads at both ends, carrying four labelled points: D, E, O and B. There's also an arrow leaving O going upwards. And another arrow leaving O going off to the right, through a point C. Now, four questions. How many points? Five. D, E, O, B and C.

9:43Which of those paths is a line? The horizontal one — the one with arrowheads at both ends. Only that one is endless in both directions. Name some rays. The arrow going up from O is one. The arrow through C from O is another. And the horizontal path gives you rays too. Ray O D goes left for ever. Ray O B goes right for ever. Name some segments. O to E. E to D. O to B. D to B. Any two labelled points on a path, with nothing endless about the bit between them.

10:20And here's an honest note about that exercise. There isn't one right answer for the last three parts. There are several defensible readings, and your book knows that. What's being marked is whether you can justify the one you gave. So here's yours. On that horizontal path with D, E, O and B — how many segments are there in total? Not how many I listed. All of them. Count carefully, and say how you knew. Comments below.

10:52Next time: what you get when two rays leave the same point — and why the corner they make is really a turn.