Problem 1 Medium Difficulty

Determine whether each statement is true or false in $ \mathbb{R}^3 $.

(a) Two lines parallel to a third line are parallel.
(b) Two lines perpendicular to a third line are parallel.
(c) Two planes parallel to a third plane are parallel.
(d) Two planes perpendicular to a third plane are parallel.
(e) Two lines parallel to a plane are parallel.
(f) Two lines perpendicular to a plane are parallel.
(g) Two planes parallel to a line are parallel.
(h) Two planes perpendicular to a line are parallel.
(i) Two planes either intersect or are parallel.
(j) Two lines either intersect or are parallel.
(k) A plane and a line either intersect or are parallel.

Answer

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04:32

WZ

Wen Z.

23:13

Chris T.

29:04

SB

Sriparna B.

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Video Transcript

Okay. We're supposed to decide if these are true or false in R. three and R three just means in space. Um Hopefully I got them all copied. Right I copied pretty fast. All right. two lines parallel to a third line are parallel to each other so This will be the this will be the 3rd line. So is there any way we could draw two lines that are parallel to it that are not parallel to each other? Yeah I think that one is true. Two lines perpendicular to a third line are parallel. Okay here's the third line. There's one line perpendicular to it. It's flat on the screen. Here's another line perpendicular to it coming out of the screen. Can they are both perpendicular but they are not parallel to each other so that one is false. Two planes parallel to a third plane are paralleled. Okay Okay let's see Here will be the third plane. So two lines that two planes that are parallel to it. Here's one here's one I'm going to say that one's true. Two planes perpendicular to a third plane are parallel. Okay here's the third plane. So I want to make a plane perpendicular to it. Here's one And then here's one. Okay so like the corner of the room so that one's false. Two lines parallel to a plane are parallel to each other. All right here's our plane. two lines parallel to it. OK here's one. Um And then here's another one underneath it. Okay they're both parallel because they never intersect it but they are not parallel to each other. Yes that was false. two lines perpendicular to a plane are parallel to each other. All right. Here's the plane. Okay. This line is perpendicular because it's going straight. Uh Let's see. Is there any way we could draw another line that is perpendicular to it? That is not parallel to that one. No, because it's going to have to make a 90° angle here. It could be on the bottom, but they are still parallel to each other. That's true. two planes parallel to align our parallel. Okay, here's the line. Um This plane is parallel to it because it never intersects it. This plane parallel to it because it never intersex. Are those two planes parallel? No, that's false. Two planes perpendicular to a line are parallel. Okay, here's the line. two planes perpendicular. Okay, so let's see, perpendicular playing. He has to make a 90° angle. Okay, sweet. Here's one here. So this is like uh the the wall of your room and then this is like a light upon the ceiling. one of those long lights. Um Okay. Can we draw another plane that's perpendicular to the light? That is not parallel to that one. Um hmm. I don't I can't think of any way. Okay. I think Has to be 90° all the way around. Okay, so, I think that one's true. two planes either intersect or are parallel. Okay, so here's a plane. Here's another plane. They intersect. Here's a plane. Here's another plane they're parallel. Is there any other way I could draw two planes? No that's true. two lines either intersect or parallel. That's false because um they could be skew, here's the picture you usually draw. Here's a box so here's a line and then here's a line and they're never going to intersect each other but they are not parallel. That's false. A plane in a line either intersect or or parallel. Here's a plane. Okay if I just randomly draw a line it's gonna intersect or it's going to be parallel. I say that one's true. All right.