Geometry

# Introduction to Euclidean geometry

## Intro to lines, line segments, and rays

Let's get familiar with the difference between lines, line segments, and rays. Hint: a ray is somewhere between a line and a line segment!
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## Intro to lines, line segments, and rays

Discussion and questions for this video
would an infinite line and an infinite ray be equally long? That's my question.
Yes. It's a tricky concept because it feels like an infinitely long ray is only half as long as an infinitely long line. But technically, half of infinity is still infinity.
So then what is the point of infinite ray and infinite line?
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A ray begins at a point and then continues on in one direction like the rays of the sun. An infinite line goes off in both directions. It has no beginning and no ending.... much like a highway has many points you can stop along the way but seems to continue on and on in both directions.
What is a segment?
A segment(when talking about a line) is only part of a line. Like he said, a line goes on forever but a line segment doesn't. ♥
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So regarding the infinite line an infinite ray concept I have a question. If a ray goes on forever it would be regarded as infinite. However a ray has a starting point unlike the line. The line is truly infinite because it goes on forever in both directions. The ray is infinite but only in one direction so wouldn't that mean that it isn't truly infinite? At some point along a infinite ray you get to starting point, at some point along an infinite line you never get to a starting point. That line has an always will exist, but the ray has a beginning.
I like this question because I believe that it invokes a deeper question about what "infinite" actually is. Before I get into my big rant about the concept of "infinite", I want to say that I believe that ray should be considered as truly infinite. Maybe my big rant will give some insight as to why I believe that.

To begin my rant, I shall compare rays and lines to number lines.

A ray is like the number line, but back when you only knew about non-negative numbers. It started at 0, and it started counting (1, 2, 3, 4, ...). We know that this process of counting can literally take forever as the number can always keep getting larger no matter how far you've gotten. So we can say that these numbers count on to infinity.

A line is like the number line after you learned about negative numbers. So, we still have the positive infinity, but the numbers can also count backwards from 0 (-1, -2, -3, -4, ...). Similar argument as before, we could take this back to negative infinity.

So, there are an infinite number of numbers greater than zero, and there are an infinite number of numbers less than zero. And there are thus an infinite number of numbers altogether. So to relate this back to rays and lines. The "positive ray" of the number line is infinite. The "negative ray" of the number line is infinite. And the entire "number line" is also infinite.

Since the number line is made up of 2 rays, does this mean that the line is twice as infinite as a ray? And if so, what does that even mean?

There is also another perspective on this concept of "infinite". I could even talk about a line segment. On the number line, we could look at only the segment of the number line between the numbers 1 and 2. We also learn at some point that there are numbers with fractional components. Between 1 and 2, there are a few we can identify that have a single decimal place after the decimal point (1.1, 1.2, 1.3, 1.4, ... all the way to 1.9). But there are still more in between those numbers, such as in between 1.3 and 1.4, there are 1.31, 1.32, 1.33, 1.34, ... up to 1.39. And this is happening between other points as well. As we do this, we are identifying numbers that are closer and closer to each other, but no matter how close they get, there are always still more points in between. And it shows no sign of slowing down. This process can be done forever, and we end up saying that there are an infinite number of numbers between 1 and 2. This is like saying that the number of points on a line segment is infinite.

A number line has an infinite number of numbers on it. We could say that in between each pair of consecutive numbers, there is a line segment. And we can conclude that there are in infinite number of line segments on this line. If each line segment has a number of points equal to infinity, and there are an infinite number of line segments, then the number of points on a line is one infinity multiplied by another infinity.

So I've just identified different ways of saying just how large the infinite-ness of a line is. So what does this tell us about what infinity is? It is not just some unimaginably large number. "infinte" is a word used to describe a process that can never be finished. There will always be more steps in the process to complete no matter how much progress you have made. Counting is a process that will never end. No matter where you stop, there is always further that you could count.
can a line also be a ray?
Lines cannot be rays.
Line: Extends forever in two directions
Ray: Extends forever in one direction and has one endpoint
Line Segment: Has two endpoints and doesn't extend forever
Would time be described as a line or a ray?
Or can it even be described that way?
Are there similar properties to ideas with more dimensions?
That is an interesting thought. To answer these question we would need to know if time had a start as well if time will have an end.
Assuming that time can be described this way, I believe that time could be represented as a ray. I'm assuming that time started 13.7 billion years ago when the big bang occurred. I am also assuming that the universe will never end which is the prevailing theory as of now, but some scientist disagree and argue that the universe could have an end.

Things can get very confusing when you take into effect the theory of relativity which states that time is not the same for everyone. If someone is moving then time is passing by slower than someone not moving! This means if i move for a little and then stand still, my experience of time will be completely different from someone standing still the entire time! (The difference in time would only be recognizable when moving at a large fraction of the speed of light which is about 300,000,000 meters per second or 186,000 miles per second so if I move the time dilation will be so small no one will recognize it but it will have occurred!) This means if I drew a time ray mine would be different than the other person not moving!

Things get very interesting if we consider extra dimensions. We live in a universe with 3 spatial dimensions and 1 time dimension. Since time is a dimension what if there was multiple time dimensions in another universe? If time had 2 dimensions time would have to be represented as an plane! This place could have an infinite edge and an non-infinite edge! What if time was 3 dimensional! Now I'm all excited! It could be any object that is three dimensional. We could have a universe with its time, if graphed, to look like anything.

Just remember Leo that even the most simplest of question can have the most mind-blowing answers.

”I don't know anything, but I do know that everything is interesting if you go into it deeply enough.”
-Richard Feynman
At 1:34, what does Sal mean by "abstract notions?"
*Abstract Notions*: Things that different/exotic, and don't usually appear in everyday life. Abstract notions are mainly used in math.
In this video, the abstract notion is the line: it continues forever

Hope that helps!
what are skew lines
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In other words, you can only have skew lines in 3 or more dimensions, because in 2D plane geometry, lines are either parallel or intersecting, but not both and not neither.
what is the difference between a ray segment and a line?
A ray has one endpoint and travels in one direction, while a line is the total opposite of a line segment, having no endpoints and travelling in 2 directions, going into infinity.
do rays relate to line segments?
it does, because you can the the infinite line in half
If there is only one point defined in a ray, could you define it? If so, how?
A ●────────>
You can't with just one point. You would need a second point somewhere on the way to show what direction it is going in.
What is the diferense between a line segement and a line and ray?
A line segment has two endpoints.
A ray has one endpoint — the other end goes off to infinity.
A line has no endpoints — both sides go off to infinity.
Hope this helps!
if a line goes on forever what would be a line in the real world?
The world and universe itself curve, so on Earth a straight line forever just circles the world endlessly, like the moon. Straight lines go on forever in terms like that as a way to explain things and because there isn't a way to make a straight line that you need to consider the curvature of the earth in everyday life. (There are a few manmade objects that would, Great wall of china for example)
at 1:49 in the video, it said that a ray had a well defined starting point? What did they mean? Did they mean the little arrow on the end?
the little arrow tells us that it can extend in that direction where that arrow points to infinity. However, the dot/ point on the opposite end of the ray (or on the opposite of the arrow ) indicates the starting point. It is fixed and cannot be moved/ extended.
Can a ray be named after a single point(origin) or is it necessary that it should pass through another point, also if a ray is passing through three points or more
for example : a ray starting at point A passing through point B further through point C and the ray is named as Ray AB can it also be called Ray BC?
A ray needs to have two points. Either name is acceptable.
if a a ray is infinte but with a starting point does it never come back onto itself? essntialy then turning itself into a line?
this concept is more the fact that it would continuously move through infinitesimal space and time and is a universal concept. you don't see the beginning or end. a ray is a point in witch a line is separated based on intended direction. because space-time is a three dimensional plane a ray can travel anywhere on this plane as long as it doesn't curve
Is a vector similar to a ray?
They are really only similar in notation. A ray is infinite, while a vector is more like a line segment with a directional component.
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would an infinite line and an infinite ray be equally long?
Technically yes, because they both have an infinite distance.
Infinite = Infinite
Where are there lines in real life? Doesn't everything have an end?
All of math is an abstract concept. It does not exist in the real world. We designed math to be highly useful in the real world, but it is just an abstract concept.

Lines do not exist any more than the number 2 exists in the real world. They are both exceedingly useful concepts for interacting with reality, but they do not literally exist.