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| 2024-10-09 22:19:15 +0200 | weary-traveler | (~user@user/user363627) (Remote host closed the connection) |
| 2024-10-09 22:10:58 +0200 | tromp | (~textual@92-110-219-57.cable.dynamic.v4.ziggo.nl) |
| 2024-10-09 22:08:10 +0200 | seydar | (~seydar@38-73-249-43.starry-inc.net) seydar |
| 2024-10-09 22:01:46 +0200 | AlexNoo__ | AlexNoo |
| 2024-10-09 21:57:31 +0200 | ljdarj1 | ljdarj |
| 2024-10-09 21:57:30 +0200 | ljdarj | (~Thunderbi@user/ljdarj) (Ping timeout: 265 seconds) |
| 2024-10-09 21:57:25 +0200 | ash3en | (~Thunderbi@ip1f10cbd6.dynamic.kabel-deutschland.de) (Quit: ash3en) |
| 2024-10-09 21:53:25 +0200 | ljdarj1 | (~Thunderbi@user/ljdarj) ljdarj |
| 2024-10-09 21:47:08 +0200 | ash3en | (~Thunderbi@ip1f10cbd6.dynamic.kabel-deutschland.de) ash3en |
| 2024-10-09 21:37:09 +0200 | tromp | (~textual@92-110-219-57.cable.dynamic.v4.ziggo.nl) (Quit: My iMac has gone to sleep. ZZZzzz…) |
| 2024-10-09 21:33:32 +0200 | gorignak | (~gorignak@user/gorignak) gorignak |
| 2024-10-09 21:33:02 +0200 | gorignak | (~gorignak@user/gorignak) (Quit: quit) |
| 2024-10-09 21:29:27 +0200 | CiaoSen | (~Jura@2a05:5800:2e5:2400:ca4b:d6ff:fec1:99da) (Ping timeout: 246 seconds) |
| 2024-10-09 21:22:03 +0200 | tromp | (~textual@92-110-219-57.cable.dynamic.v4.ziggo.nl) |
| 2024-10-09 21:21:54 +0200 | AlexZenon | (~alzenon@178.34.163.165) |
| 2024-10-09 21:19:13 +0200 | <tomsmeding> | I guess the scaling is superfluous, but it helps intuition |
| 2024-10-09 21:18:49 +0200 | <tomsmeding> | "so that B-A is (1,0)" -> "so that B-A is (1,0,0)" |
| 2024-10-09 21:17:24 +0200 | tromp | (~textual@92-110-219-57.cable.dynamic.v4.ziggo.nl) (Quit: My iMac has gone to sleep. ZZZzzz…) |
| 2024-10-09 21:17:02 +0200 | <tomsmeding> | convoluted, though |
| 2024-10-09 21:16:36 +0200 | <tomsmeding> | that feels like an argument |
| 2024-10-09 21:16:23 +0200 | <tomsmeding> | and angles in the projection are preserved by rotations around the z-axis and scalings |
| 2024-10-09 21:16:06 +0200 | <tomsmeding> | it's not parallel to the z-axis because that would mean that OAB is parallel to the x-y plane, which it isn't, so it has a nonzero y component |
| 2024-10-09 21:15:34 +0200 | <tomsmeding> | ah: B-A lies in the x-y plane, so rotate and scale the whole system so that B-A is (1,0); then the (rotated) cross product is still orthogonal, so it's in the y-z plane |
| 2024-10-09 21:15:22 +0200 | briandaed | (~root@185.234.210.211.r.toneticgroup.pl) (Remote host closed the connection) |
| 2024-10-09 21:14:28 +0200 | <tomsmeding> | it is relevant that B-A lies in the x-y plane, because without that restriction, it's easy to think of two orthogonal vectors in R^3 that are not orthogonal after projection to the x-y plane |
| 2024-10-09 21:14:24 +0200 | AlexNoo_ | (~AlexNoo@178.34.163.165) (Ping timeout: 246 seconds) |
| 2024-10-09 21:13:30 +0200 | <dolio> | Yeah, I'm not sure how to see geometrically that the projection is still orthogonal. |
| 2024-10-09 21:13:14 +0200 | weary-traveler | (~user@user/user363627) user363627 |
| 2024-10-09 21:13:00 +0200 | <tomsmeding> | but that's cheating again |
| 2024-10-09 21:12:53 +0200 | <tomsmeding> | you can do that algebraically (the cross product is orthogonal to B-A, but the z-component of B-A is zero, hence it's orthogonal regardless of the z component of the cross product, hence also at zero, hence also when projected to the x-y plane) |
| 2024-10-09 21:12:16 +0200 | AlexNoo | (~AlexNoo@178.34.151.120) (Ping timeout: 252 seconds) |
| 2024-10-09 21:11:35 +0200 | <tomsmeding> | right |
| 2024-10-09 21:11:32 +0200 | AlexNoo__ | (~AlexNoo@178.34.163.165) |
| 2024-10-09 21:11:14 +0200 | AlexZenon | (~alzenon@178.34.151.120) (Ping timeout: 260 seconds) |
| 2024-10-09 21:11:11 +0200 | <tomsmeding> | it was a \qed with a ? inside |
| 2024-10-09 21:11:06 +0200 | <tomsmeding> | when I did a bachelor in maths I had some fun with a fellow student and wrote a macro which, translated to English, would I guess be \maybeBoxy |
| 2024-10-09 21:10:48 +0200 | wootehfoot | (~wootehfoo@user/wootehfoot) (Read error: Connection reset by peer) |
| 2024-10-09 21:10:44 +0200 | <dolio> | Well, you need some fact about projections back to the x-y plane. |
| 2024-10-09 21:10:12 +0200 | <tomsmeding> | \qed? |
| 2024-10-09 21:10:09 +0200 | <tomsmeding> | the cross product is normal to the plane spun up by the triangle OAB, hence it is normal to all vectors in that plane, in particular B-A |
| 2024-10-09 21:09:40 +0200 | <tomsmeding> | okay fair |
| 2024-10-09 21:09:30 +0200 | <tomsmeding> | well it always was |
| 2024-10-09 21:09:25 +0200 | <tomsmeding> | yes ... oh |
| 2024-10-09 21:09:08 +0200 | <dolio> | It's normal to a plane containing the line between the a and b points. |
| 2024-10-09 21:08:39 +0200 | AlexNoo_ | (~AlexNoo@178.34.163.165) |
| 2024-10-09 21:08:38 +0200 | <tomsmeding> | so I'm not sure about your "contains the line you care about" |
| 2024-10-09 21:07:55 +0200 | <tomsmeding> | the cross product is going to be _normal_ to that plane |
| 2024-10-09 21:07:45 +0200 | <tomsmeding> | that's just the same plane as we've been looking at before ("the triangle OAB"), only shifted down by z :p |
| 2024-10-09 21:07:24 +0200 | <dolio> | Yes. |
| 2024-10-09 21:07:17 +0200 | <tomsmeding> | and "the plane" is the one generated by {(a1,a2,0), (b1,b2,0), (0,0,-z)}? |