turbid bear
I am learning geometry with a "hard" book, which i do not mind I find it fun. But this question is very confusing
From what I know. Two distinct straight lines can intersect only once, and if a line passes through 2 points of a plane, then all points of the line lie on that plane.
I may be dumb but I am just confused, only thing I can think of is a plane where the the line outside of it intersects the edge (like touching the edge of a cube) other than that im stumped.
latent vergeBOT
narrow nymph
There's two things that can happen with a line that does not lie in a given plane.
Either it's parallel to the plane and they never intersect, or the line pierces the plane in one point.
They're essentially asking you to show this.
From
if a line passes through 2 points of a plane, then all points of the line lie on that plane.
you should be able to argue by contradiction.
turbid bear
Im really just diving into math so bare with me, what exactly am I supposed to show, or rather how do I show? proof? drawing?
narrow nymph
Are you familiar with proofs by contradiction?
turbid bear
Only vaguely, my understanding is you prove the opposite case, and in doing so you prove your case?
Show that this cannot be false, and so it must be true
narrow nymph
Yep.
So this question is asking you to show that
If you have a plane and a line not contained in the plane, then the line intersect the plane at at most 1 point.
For contradiction you assume the premise, so you assume you're given a plane and a line not on the plane, and you suppose the opposite conclusion holds
What would be the opposite conclusion?
turbid bear
It would be it can insersect at more than one point or none?
narrow nymph
Yes, more than one point of intersection.
So now you have a plane, a line not on the plane, and more than one point of intersection.
You have to argue this is not possible.
Why?
turbid bear
Because if a line passes through more than one point in a plane it has to be inside of the plane?
narrow nymph
Indeed. If you have more than one point of intersection, then you have at least 2. So your line shares 2 points with the plane and therefore lies in the plane. This is in contradiction with the fact that the line was specifically chosen not on the plane.
turbid bear
Right the line was chosen not on the plane, but and so its not possible for it to have more than one point of intersection because then it would be in the plane. So the line would be in the plane and not in the plane and thats not possible..
right?
narrow nymph
Yes
turbid bear
I understand okay, thank you so much lol
turbid bear
Yes
Big problem was me visualizing it in only 2 dimensions, so I when drawing it I saw a line going OVER a plane multiple times rather than piercing through it, if that makes sense
narrow nymph
Ohh yeah that wouldn't work
latent vergeBOT
I understand okay, thank you so much lol
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turbid bear
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