Understanding Parallel Lines: Definition and Differences

[nateja]

Homework Statement

The question states True or False: Two lines parallel to a third line are parallel

Homework Equations

You need to know the difference between skew, parallel and perpendicular

The Attempt at a Solution

I thought of three parallel planes (which have an infinite amount of lines) and those lines never cross as long as they stay on their separate planes. But do parallel lines have to be in the same direction?

 

[Tsunoyukami]

Yes. If line A is parallel to line B and line B is parallel to line C then line A must be parallel to line C, essentially by definition of being parallel.

 

[nateja]

I have a few more True or False questions:

Two planes perpendicular to a third plane are parallel? I said T because I thought the same plane would be considered parallel - like with vectors. But I feel like this is considered an intersection.

Two lines perpendicular to a plane are parallel? I said false because the lines could be orientated in different directions

Two planes parallel to a line are parallel? I said true because planes have no direction and as long as they don't intersect they should be parallel.

A plane and a line eithere intersect or are parallel? I guessed this because I was thinking skew lines... but I see my error here.

 

[nateja]

Are two lines parallel if they lie on parallel planes?? I'm starting to see the difference here between parallel and skew.

I hope I'm not asking too many questions... it just helps to think it out sometimes.

 

[Tsunoyukami]

1) Sounds good.
2) How can these two lines be perpendicular to a plane in different directions? Can you explain?
3) Sounds good.
4) What is your "final answer" then?

And from your next post:

5) If you have two planes that are parallel oriented standing up (imagine the two covers of a book) you can draw a line on one cover going straight up and down and one on the other cover going directly across, so this is false because these two lines would not be parallel.No such thing as asking too many questions! It's the best way to learn (as long as you're thinking about it ;) ).

 

[nateja]

Ok... Thanks for the number 5 clarification. I now understand that parallel means not touching and same direction.

So my guess for number 4 is that it is True. A line will intersect with a plan always - unless it is parallel.

Number two was a misunderstanding on my part. I was thinking of a vector equation of a line. And I thought that if you use opposite directional vectors, then the lines would not be parallel. But they will be parallel because you can factor out the negative. So number 2 is True.



I didn't directly state this, but for number 1 and number 3, my explanation is wrong. the correct answer (according to the book) is that number 1 and 3 are false. So I'm trying to understand why...

I would think that number 1 would be true, but it is false because the normal vectors could be in different directions... and I believe this is part of the definition of parallel planes (I'll get back to you... have to double check in the book)

So if that definition is of parallel planes is true, then number 3 is false by the same logic.

 

[nateja]

Ok, I'm right with the last 3! I reread the planes section and it states "two planes are parallel if their normal vectors are parallel"... which is confusing now because I realize that if I factor out a negative for two opposite directional vectors then I'll get the same vector... same as a directional vector in a line.

Does that make sense? I'm confused on how number 1 and 3 are not true... the vectors v and -v are parallel, right?

 

[Tsunoyukami]

I also thought that 1 and 3 would be true.

I have to double check the formula of a plane and of the normal vector of a plane before I can give a "proper" response.

I personally considered v and -v to be parallel, but maybe there is a more precise definition provided in your textbook. Not sure...EDIT: I reread your post and 3 is false.

If you have a line you can have a plane parallel to it and another plane parallel to it such that these two planes will intersect. Try imagining it - hard to explain using words aha.

 

[nateja]

That makes sense for number 3... kind of like 'skew' planes is how I"m thinking of it.