WEBVTT 00:00:06.785 --> 00:00:07.410 JOEL LEWIS: Hi. 00:00:07.410 --> 00:00:08.720 Welcome to recitation. 00:00:08.720 --> 00:00:11.200 In lecture, you started learning about vectors. 00:00:11.200 --> 00:00:13.050 Now vectors are going to be really important 00:00:13.050 --> 00:00:15.360 throughout the whole of this course. 00:00:15.360 --> 00:00:16.860 And I wanted to give you one problem 00:00:16.860 --> 00:00:19.211 just to work with them in a slightly different context 00:00:19.211 --> 00:00:20.960 than what we're going to do in the future. 00:00:20.960 --> 00:00:23.010 So this is the context of Euclidean geometry. 00:00:23.010 --> 00:00:27.359 So some of you have probably seen this problem 00:00:27.359 --> 00:00:29.150 that we're going to solve, but you probably 00:00:29.150 --> 00:00:30.910 haven't seen it solved with vectors. 00:00:30.910 --> 00:00:32.470 So let's take a look at it. 00:00:32.470 --> 00:00:34.260 So what I'd like you to do is show 00:00:34.260 --> 00:00:37.590 that the three medians of a triangle intersect at a point, 00:00:37.590 --> 00:00:40.370 and the point is 2/3 of the way from each vertex. 00:00:40.370 --> 00:00:42.780 So let me just remind you of some terminology. 00:00:42.780 --> 00:00:45.870 So in a triangle, a median is the segment 00:00:45.870 --> 00:00:49.890 that connects one vertex to the midpoint of the opposite side. 00:00:49.890 --> 00:00:55.761 So here, this point M is exactly halfway between B and C. 00:00:55.761 --> 00:00:56.260 So OK. 00:00:56.260 --> 00:00:58.360 So every triangle has three medians-- 00:00:58.360 --> 00:01:01.100 one from each vertex connected to the midpoint 00:01:01.100 --> 00:01:03.660 of the opposite side-- and what I'm asking you to show 00:01:03.660 --> 00:01:07.850 is that these three medians all intersect in the same point. 00:01:07.850 --> 00:01:10.950 And also, that this point divides the median 00:01:10.950 --> 00:01:14.390 into two pieces, and the big piece is twice 00:01:14.390 --> 00:01:15.970 as large as the small piece. 00:01:15.970 --> 00:01:20.530 So this is 2/3 of the median, and this is 1/3 of the median. 00:01:20.530 --> 00:01:23.520 So, why don't you take a few minutes, work that out-- try 00:01:23.520 --> 00:01:28.215 and do it using vectors as much as possible-- pause the video, 00:01:28.215 --> 00:01:29.965 come back, and we can work on it together. 00:01:39.080 --> 00:01:42.320 So hopefully you had some luck working on this problem. 00:01:42.320 --> 00:01:44.460 Let's get started on it. 00:01:44.460 --> 00:01:48.020 So to start, I actually want to rephrase 00:01:48.020 --> 00:01:50.110 the question a little bit. 00:01:50.110 --> 00:01:53.680 And I'll rephrase it to an equivalent question 00:01:53.680 --> 00:01:56.210 that's a little bit more clear about how 00:01:56.210 --> 00:01:57.250 we want to get started. 00:01:57.250 --> 00:02:00.140 So another way to say this problem is 00:02:00.140 --> 00:02:03.430 that it's asking us to show-- so for each median, 00:02:03.430 --> 00:02:08.110 say this median AM here, where M is the midpoint of side BC, 00:02:08.110 --> 00:02:11.220 there exists a point on the median that divides it 00:02:11.220 --> 00:02:15.120 into a 2:1 ratio, so the point that's 2/3 from the vertex 00:02:15.120 --> 00:02:16.960 to the midpoint of the opposite side. 00:02:16.960 --> 00:02:20.160 So for example, you know, there's a point-- so, 00:02:20.160 --> 00:02:25.330 let's call it P, say, at first-- so there's a point P 00:02:25.330 --> 00:02:27.930 such that AP is twice PM. 00:02:27.930 --> 00:02:28.430 OK? 00:02:28.430 --> 00:02:30.520 And similarly, there's some point-- maybe called 00:02:30.520 --> 00:02:36.110 Q-- that's 2/3 of the way from B to the midpoint of this side. 00:02:36.110 --> 00:02:39.030 And there's some point that's 2/3 of the way from C 00:02:39.030 --> 00:02:40.450 to the midpoint of this side. 00:02:40.450 --> 00:02:43.240 And so an equivalent formulation of the question 00:02:43.240 --> 00:02:45.440 is to show that these three points are really 00:02:45.440 --> 00:02:46.890 the same point. 00:02:46.890 --> 00:02:48.880 That they're all in the same place. 00:02:48.880 --> 00:02:52.360 So one way we can do that is that we 00:02:52.360 --> 00:02:55.590 can compare the position vectors of those three points. 00:02:55.590 --> 00:02:58.490 And if those three points all have the same position vector, 00:02:58.490 --> 00:03:00.800 then they're all in exactly the same position. 00:03:00.800 --> 00:03:03.689 So in order to do that we need some origin. 00:03:03.689 --> 00:03:05.230 And it happens that for this problem, 00:03:05.230 --> 00:03:07.405 it doesn't matter where the origin is, 00:03:07.405 --> 00:03:09.030 and so I'm not going to draw an origin, 00:03:09.030 --> 00:03:10.770 but I'm going to call it O. 00:03:10.770 --> 00:03:14.490 So we're going to set up a vector coordinate system 00:03:14.490 --> 00:03:20.240 with origin O. And now I want to look 00:03:20.240 --> 00:03:26.970 at what the vector from O to P is in terms of the vectors 00:03:26.970 --> 00:03:29.420 connecting O to A, B, and C. Right? 00:03:29.420 --> 00:03:32.160 Those are the vectors that determine 00:03:32.160 --> 00:03:33.530 the vertices of the triangle. 00:03:33.530 --> 00:03:36.190 And so I want to relate the location of P 00:03:36.190 --> 00:03:38.620 to the locations of A, B, and C. 00:03:38.620 --> 00:03:42.330 So the first thing to do is that-- well, in order 00:03:42.330 --> 00:03:46.000 to talk about where P is, I know how P is related to A and M 00:03:46.000 --> 00:03:48.770 and I know how M is related to B and C. 00:03:48.770 --> 00:03:51.580 So let's first figure out what the position vector of M 00:03:51.580 --> 00:03:54.337 is in terms of the position vectors of A, B, and C, 00:03:54.337 --> 00:03:55.920 and then we can use that to figure out 00:03:55.920 --> 00:04:03.240 the position vector of P. 00:04:03.240 --> 00:04:10.260 So M is the midpoint of the segment BC. 00:04:13.110 --> 00:04:15.670 So I think we saw this in lecture. 00:04:15.670 --> 00:04:22.232 What this means is that the position vector OM is exactly 00:04:22.232 --> 00:04:25.770 the average of the position vectors of B and C. 00:04:25.770 --> 00:04:31.430 It's 1/2 of the quantity OB plus OC. 00:04:31.430 --> 00:04:31.930 All right? 00:04:31.930 --> 00:04:35.470 So it's easy to express the position 00:04:35.470 --> 00:04:38.300 vector of the midpoint of a segment in terms 00:04:38.300 --> 00:04:40.280 of the position vectors of the endpoints. 00:04:40.280 --> 00:04:42.450 You just add the position vectors of the endpoints 00:04:42.450 --> 00:04:43.970 and divide by 2. 00:04:43.970 --> 00:04:46.920 So if you like, this is equivalent 00:04:46.920 --> 00:04:49.910 to the geometric fact that the diagonals of a parallelogram 00:04:49.910 --> 00:04:51.470 bisect each other. 00:04:51.470 --> 00:04:55.340 So that's the position vector of M. 00:04:55.340 --> 00:04:59.490 Now we have to figure out what the position vector of P is. 00:04:59.490 --> 00:05:03.930 So in order to do this we can note, 00:05:03.930 --> 00:05:06.560 that in order to get from the origin to point P, 00:05:06.560 --> 00:05:09.700 well, what we have to do is we have to go from the origin-- 00:05:09.700 --> 00:05:11.460 wherever it is-- to A, and then we 00:05:11.460 --> 00:05:15.720 have to go from A 2/3 of the way to M. All right? 00:05:15.720 --> 00:05:29.440 So the vector OP is equal to OA plus 2/3 of the vector AM. 00:05:29.440 --> 00:05:29.940 Right? 00:05:29.940 --> 00:05:31.530 Because we go 2/3 of the way from A 00:05:31.530 --> 00:05:33.730 to M in order to get from A to P. 00:05:33.730 --> 00:05:36.570 This is because we've chosen P to be the point that 00:05:36.570 --> 00:05:42.390 divides segment AM into a 2:1 ratio so that AP is 2/3 of AM. 00:05:42.390 --> 00:05:43.000 OK. 00:05:43.000 --> 00:05:43.590 So good. 00:05:43.590 --> 00:05:46.030 So now we need the vector AM. 00:05:46.030 --> 00:05:48.970 Well, we know what the position vector of A is. 00:05:48.970 --> 00:05:50.190 It's just OA. 00:05:50.190 --> 00:05:53.500 And we also know what the position vector of M is. 00:05:53.500 --> 00:05:54.500 It's OM. 00:05:54.500 --> 00:06:01.880 So that means that AM is just the difference of those two 00:06:01.880 --> 00:06:02.870 vectors. 00:06:02.870 --> 00:06:09.360 It's going to be OM minus OA. 00:06:09.360 --> 00:06:12.520 Another way to say this is that if you add OA to both sides, 00:06:12.520 --> 00:06:16.120 you have that OA plus AM equals OM. 00:06:16.120 --> 00:06:18.160 In other words, to go from O to M, 00:06:18.160 --> 00:06:20.510 first you can go from O to A, and then go from A to M. 00:06:20.510 --> 00:06:21.340 All right. 00:06:21.340 --> 00:06:24.410 And I've just subtracted OA onto the other side here. 00:06:24.410 --> 00:06:28.640 So we can write AM in terms of OM and OA. 00:06:28.640 --> 00:06:30.440 And we also-- we have an expression for OM 00:06:30.440 --> 00:06:33.290 here in terms of OB and OC. 00:06:33.290 --> 00:06:36.190 So that means we can get an expression for AM 00:06:36.190 --> 00:06:39.120 in terms of OA, OB, and OC. 00:06:39.120 --> 00:06:39.830 So let's do that. 00:06:39.830 --> 00:06:43.000 So that's just by substituting from here into here. 00:06:43.000 --> 00:06:47.960 So if I do that, I get that AM is 00:06:47.960 --> 00:06:58.550 equal to-- so OM is 1/2 OB plus 1/2 OC, 00:06:58.550 --> 00:07:01.090 and now I just subtract OA. 00:07:03.685 --> 00:07:04.510 All right. 00:07:04.510 --> 00:07:08.790 So that's what AM is, putting these two equations together. 00:07:08.790 --> 00:07:10.830 I get that that's AM. 00:07:10.830 --> 00:07:13.100 And so now I need to figure out what OP is. 00:07:13.100 --> 00:07:14.980 So for OP, I just need to substitute 00:07:14.980 --> 00:07:18.170 in this new expression that I've got for AM. 00:07:18.170 --> 00:07:21.444 So I have OP is equal to, well it's 00:07:21.444 --> 00:07:28.542 equal to OA plus 2/3 of what I've written 00:07:28.542 --> 00:07:30.860 just right above-- 2/3 of AM. 00:07:30.860 --> 00:07:40.500 So that's 1/2 OB plus 1/2 OC minus OA. 00:07:40.500 --> 00:07:41.020 OK. 00:07:41.020 --> 00:07:43.760 And so now you can multiply this 2/3 in-- you know, 00:07:43.760 --> 00:07:46.710 just distribute the scalar multiplication 00:07:46.710 --> 00:07:49.300 across the addition there-- and then we can rearrange. 00:07:49.300 --> 00:07:52.890 We'll have two terms involving OA and we can combine them. 00:07:52.890 --> 00:07:58.280 So we'll see we have a plus OA minus 2/3 OA. 00:07:58.280 --> 00:08:03.450 So that's going to be equal to 1/3 OA. 00:08:03.450 --> 00:08:07.280 And then we have, OK so 2/3 times 1/2 OB. 00:08:07.280 --> 00:08:16.730 So that's plus 1/3 OB plus 1/3 OC. 00:08:16.730 --> 00:08:19.160 So this gives us a simple formula 00:08:19.160 --> 00:08:24.050 for the position vector of P-- that vector OP-- in terms 00:08:24.050 --> 00:08:25.980 of the position vectors of A, B, and C. 00:08:25.980 --> 00:08:28.040 So in particular, it's actually-- 00:08:28.040 --> 00:08:32.920 because P is the special point-- it's 1/3 of their sum. 00:08:32.920 --> 00:08:35.830 Of the sum OA plus OB plus OC. 00:08:35.830 --> 00:08:38.010 OK, so that's where P is. 00:08:38.010 --> 00:08:39.680 Now to finish the problem, I just 00:08:39.680 --> 00:08:43.460 have to show that this is the same location as the point that 00:08:43.460 --> 00:08:45.470 trisects the other medians. 00:08:45.470 --> 00:08:46.940 So how would I do that? 00:08:46.940 --> 00:08:49.380 Well, I could go back to my triangle 00:08:49.380 --> 00:08:51.912 and I could do exactly the same thing. 00:08:51.912 --> 00:08:54.120 So I could-- maybe I'll give this point a name, also. 00:08:54.120 --> 00:08:56.770 I'll call this midpoint N, say. 00:08:56.770 --> 00:09:01.890 So I could let Q be the point that lies 2/3 of the way 00:09:01.890 --> 00:09:04.540 from B to N. And then I could write down 00:09:04.540 --> 00:09:08.277 the position vector of N in terms of OA, OB, and OC. 00:09:08.277 --> 00:09:10.860 And then I can use that to write down the position vector of Q 00:09:10.860 --> 00:09:14.580 in terms of OA, OB, and OC, and I'll get some expression. 00:09:14.580 --> 00:09:16.390 And what will happen at the end-- 00:09:16.390 --> 00:09:18.720 I hope if I'm lucky-- that expression 00:09:18.720 --> 00:09:22.220 will be equal to this expression that I found over here. 00:09:22.220 --> 00:09:22.760 OK? 00:09:22.760 --> 00:09:24.950 So you can go through and do that, 00:09:24.950 --> 00:09:28.210 and if you do that, what you'll find is that in fact it works. 00:09:28.210 --> 00:09:31.140 So there's actually a sort of clever, shorter way 00:09:31.140 --> 00:09:31.960 of seeing that. 00:09:31.960 --> 00:09:36.730 Which is that this formula is symmetric in A, B, and C. 00:09:36.730 --> 00:09:42.860 So that means if I just relabel the points A, B, and C, 00:09:42.860 --> 00:09:45.570 this expression for the position vector doesn't change. 00:09:45.570 --> 00:09:48.010 So rather than going through that process 00:09:48.010 --> 00:09:49.900 that I just described, you can also 00:09:49.900 --> 00:09:55.150 say, well, in order to look at, say Q, 00:09:55.150 --> 00:09:57.210 instead of P, what I need to do is I just 00:09:57.210 --> 00:10:00.150 need to switch B and A. I need to do exactly the same thing 00:10:00.150 --> 00:10:02.184 but the roles of A and B are interchanged. 00:10:02.184 --> 00:10:04.350 Well, if the roles of A and B are interchanged, then 00:10:04.350 --> 00:10:05.891 in the resulting formula, I just have 00:10:05.891 --> 00:10:07.570 to interchange the roles of A and B, 00:10:07.570 --> 00:10:09.960 but that won't change the value of this expression. 00:10:09.960 --> 00:10:14.170 So by symmetry, the point I get really is going to be the same. 00:10:14.170 --> 00:10:16.240 If you don't like that argument, I 00:10:16.240 --> 00:10:20.360 invite you to go through this computation 00:10:20.360 --> 00:10:24.600 again in the case of the other medians. 00:10:24.600 --> 00:10:27.860 In either case, what you'll find is that the points that trisect 00:10:27.860 --> 00:10:31.100 the three medians all have position vector 1/3 00:10:31.100 --> 00:10:34.700 OA plus 1/3 OB plus 1/3 OC, but that means 00:10:34.700 --> 00:10:36.230 they're the same point. 00:10:36.230 --> 00:10:41.260 So what we've shown then, is that the points that 00:10:41.260 --> 00:10:45.290 trisect the three medians-- that trisects, that divide them 00:10:45.290 --> 00:10:48.809 into 2:1 ratios from the vertex to the midpoint 00:10:48.809 --> 00:10:51.100 of the opposite side-- that those three points all have 00:10:51.100 --> 00:10:52.390 the same position vector. 00:10:52.390 --> 00:10:54.210 So in fact, they're the same point, 00:10:54.210 --> 00:10:56.390 and that's what we wanted to show, right? 00:10:56.390 --> 00:10:59.660 We wanted to show that there's one point that trisects 00:10:59.660 --> 00:11:01.380 all three of those medians. 00:11:01.380 --> 00:11:04.080 So we've shown that the three points that trisect them 00:11:04.080 --> 00:11:05.480 are actually the same. 00:11:05.480 --> 00:11:09.530 So that's the same conclusion, phrased differently. 00:11:09.530 --> 00:11:11.906 So I think I'll end there.