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In this video for 3D geometry, we will be talking about two topics. One is the angle bisector of two planes and one is line of intersection of two planes. Both of the topics are important especially the line of intersection of two planes, and I will be spending some time on that. Angle bisector of two planes is a very easy topic, something you should remember while preparing for JEE. I will just give you the formula and I won’t be doing a problem but I will spending sometime on this (Line of Intersection of two planes), because this is a relatively important topic. So let us say we have a pair of planes. And you have to find the angle bisector. This is the angle bisector — plane which is the angle bisector of two planes. This angle is the same as this angle. You have been given equations for P1 and P2. You have find equation of angle bisector. If you think about this — easiest way to think about planes is using a notebook. I always recommend you do that if you are a...

so thank you for coming to this evening's event public fantasy Judith Barry one of our senior fellows here at the center for advanced visual studies we at MIT and locals welcome you Judith thank you for coming Judith has been an outstanding feminist Voyer media productionist sounding board Evangelical media artist and participant writer bookmaker traveler and architect I can't say enough good things about the input that Judith has given me personally over my own artistic career born in Columbus Ohio she's been everywhere in fact I usually see Judith in far-flung cities not where she lives echo in the shadow of the city Vampira is an exod ition piece that I saw many years ago and I'm always reminded of even in my current production it's also true that many of the things Judith has said over the years appears in the writings of others later and I'm surprised but somehow we all remember Judith when she first said when she talked about the interactivit...

GILBERT STRANG: OK, ready for part three of this vision of linear algebra. So the key word in part three is orthogonal, which again means perpendicular. So we have perpendicular vectors. We can imagine those. We have something called orthogonal matrices. That's when-- I've got one here. An orthogonal matrix is when we have these columns. I'm always going to use the letter Q for an orthogonal matrix. And I look at its columns, and every column is perpendicular to every other column. So I don't just have two perpendicular vectors going like this. I have n of them because I'm in n dimensions. And you just imagine xyz axes or xyzw axes, go up to 4D for relativity, go up to 8D for string theory, 8 dimensions. We just have vectors. After all, it's just this row of numbers or a column of numbers. And we can decide when things are perpendicular by that test. Like say the test for Q1 to be perpendicular to Qn is that row times that column. When I say times,...

PROFESSOR: Let me discuss, for a second, effective potential. And-- actually, let me ask you a question to get you started. We're going to look at high n. N-- so n's, say, 100. And then l will go from 0 up to 99. Very high l's. What's going to be different from them? Well here is one possibility, and that's a possibility that happens. The orbits are going to go from being circular to being elliptical as you change l. And I ask you-- which l would correspond to circular orbit, and which l would correspond to the most elliptical orbit? So let's take a poll/ so most circular orbit-- is it l equals 0 or l equals 99? Who votes for l equals 0 for the most circular orbit? I have about half [INAUDIBLE], a little bit more than half. OK. Who votes for l equals 99? [INAUDIBLE] Which is the most [INAUDIBLE]? OK, this is funny, because we all associate most circular with l equals 0, spherically symmetric. But it's wrong. This is the most circular. AUDIENCE:...

PROFESSOR: OK, ready for more? I defined a problem for you. Now let's address it. Said that if we have one coordinate system, and if we have some vector, q, that's defined as a second-rank tensor, aij times some other vector p sub j-- and let me digress in passing. I am very careful to say "a second-rank tensor" and not a "second-order tensor," because higher-order order terms means negligible and non-important. And when I say "second-order tensor," I don't mean to say it's not important and negligible. It's very important, so I say "rank," which has some sort of dignity to it. So I don't like the term "order," because it has another meaning. OK, so here is a tensor that relates a vector pj to give us the components of a vector qi. If we change coordinate system, the components of p, representing exactly one in the same vector, wink on and off and take different values. The values for q take on diff...