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Geometry

Introduction to Euclidean geometry

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Lines, line segments, and rays

Difference between lines, line segments, and rays
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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.
What is the difference of a ray line segment and a line?
A line segment connects two points and stops at those two points.
A ray starts at a point and continues forever in one direction.
A line continues forever in both directions.
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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What if the ray was coming from both sides like this:
<--------o-------->
Would this be a ray or a line?
If it does not start with a point, then it is not a ray. it is a line.
Ray- o----------->
Line- <----------->
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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?
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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
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
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.
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At 1:34, what does Sal mean by "abstract notions?"
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*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!
If there is only one point defined in a ray, could you define it? If so, how?
A ●────────>
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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.
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do rays relate to line segments?
it does, because you can the the infinite line in half
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what is the difference between a ray segment and a line?
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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.
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?
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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?
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A ray needs to have two points. Either name is acceptable.
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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?
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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
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would an infinite line and an infinite ray be equally long?
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Technically yes, because they both have an infinite distance.
Infinite = Infinite
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Where are there lines in real life? Doesn't everything have an end?
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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.
1 Comment
At 2:33 they say that the figure is a line. But two endpoints are given, so it doesn't make sense for it to be a line and not a line segment.
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It's useful to focus on the arrows - at both ends - that tells us, "YES" it can be a line.
(In the video around 1:00 minute, when Sal sketched a line... I think he should sketch two points along the line, however, he was busy emphasizing those arrows at each end to show it is a line.)
It is good that you saw that it seems to represent a line segment, because the two points there, show a segment within the line. (If the question was, can this represent a LINE SEGMENT? it would be right to say, YES.)
A LINE is defined by two points (where any point has a location) and the line does NOT STOP at either point, so two "arrows" in the sketch always means "this is a line."
Two points A and C and the sketch extends with lines and arrows away from both of two points, can correctly be identified as line segment AC, ray AC, ray CA, and line AC (or line CA).
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how can you remember the segments more easily
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A segment is not infinite and has 2 endpoints.
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Sal says that a line basically goes on forever. So would you be able to count the equator as a line?
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You can, but not in the same way. The equator is a great circle of a sphere (assuming an idealized earth). In Euclidean (or flat space) geometry, this great circle would simply be a circle. However, if you allow yourself to deviate from the Euclidean parallel postulate, you can construct another geometry called spherical geometry in which a great circle is legitimately considered a line.
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What would be the equivalent of a line on a 3-d or 4-d plane?
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Hello,

An example for a 3 dimension element is "space", a volume, if you consider a cube, dimensions would "fit" with length, height and width.
For a 4 dimension element, usually in Physics, we add time, and in order to make this dimension change, you move in the time (like Marty Mc Fly in his Dolorean ;-) )

I hope it helps...
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How come lines have no thickness? Isn't it as thick as the line?
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When you draw a line it has thickness, but that is just a representation. The abstract idea of a line, however, does not have any thickness.
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How can a infinite line be equally to a infinite ray if the line goes on on both ends and the ray goes on on one end?
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A line can go on infinitely in both directions. A ray can only go infinitely in one direction because it starts from 1 point. Typically, the lines you see in school at this time are actually line segments. Line segments have 2 endpoints; 1 on each end.
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can a line segment be infinte cause you could place the two points anywhere? That's my question
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an infinite line is called a line or ray
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So what is the difference between a ray and a line?
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A line goes on forever in both directions. A ray has a starting point, then goes on forever in one direction, kind of like a flashlight beam.
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So Is the universe a ray,line or line segment?
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The universe is a plane. Like a plane, it goes on forever and ever.
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Is there such thing as a acute line segment
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No, "acute" only applies to angles that are smaller than 90 degrees.
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Does a line have to have arrows at the end?
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Yes it does. Both ends in fact. This is so to distinguish it from a ray, or a segment.
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would the steps to finding the answer to these problems, always be the same?
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there are lots of ways to solve problems in math. i like to use one same way each time so i dont get mixed up :)
nowadays why do people call line segments ,lines
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In a conversation outside a math classroom, "line" is descriptive enough, But math users like to be precise.
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Why would there be line segments inside a line? I thought It was only part of a line and it wouldn't be inside
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If there are four points on a line, there are nine different segments. Think of each little space between the points as a segment by itself. All the other segments create other segments, and each single segment can be used as many times as the patterns require. It's the same basic principle as those "Jim has three shirts and three pairs of pants. How many combinations are there?' questions.

Hope this helps and I didn't confuse you!
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What are the similarities and differences between a line and a line segment?
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A line segment is a portion of a line that has a finite length.
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What is the easiest way to remember the difference between line segments and lines?
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a line segment has a dot on each end and a line has an arrow on each end . also a line goes forever in both directions .
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Does it mean you can't draw a ray and line?
And why people usually call line segment as line?
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Well, you can't exactly draw one, but you can indicate them by putting arrow heads.
I think a lot of people grow up using the term line, and can't be bothered saying line segment.
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how do you get to the practice based on this video
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Sal Just a side note, if you play the video," recognizing lines"in 4th grade, and add captions, it talks about beer and counting Koreans, I think the computer takes what your saying and mishears it.
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why did they come up with signs to represent rays, line segments, and lines?
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Mathematics is a language with its own set of symbols. There are many symbols that simply make it shorter to write than to have to write out the words when designing proofs. Rays are used in the definition of angles and we have to have a concept of lines before we can take only a part of it (a line segment).