[Script Info] Title: [Events] Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text Dialogue: 0,0:00:02.26,0:00:06.91,Default,,0000,0000,0000,,The natural way to describe the\Nposition of any point is to use Dialogue: 0,0:00:06.91,0:00:10.86,Default,,0000,0000,0000,,Cartesian coordinates. In two\Ndimensions it's quite easy. Dialogue: 0,0:00:11.58,0:00:17.85,Default,,0000,0000,0000,,We just have. Picture like\Nthis and so we have an X Dialogue: 0,0:00:17.85,0:00:19.81,Default,,0000,0000,0000,,axis and Y Axis. Dialogue: 0,0:00:22.13,0:00:27.46,Default,,0000,0000,0000,,Origin oh where they cross and\Nif we want to have vectors in Dialogue: 0,0:00:27.46,0:00:31.97,Default,,0000,0000,0000,,that arrangement, what we would\Nhave is a vector I associated Dialogue: 0,0:00:31.97,0:00:37.30,Default,,0000,0000,0000,,with the X axis and a vector Jay\Nassociated with the Y axis. Dialogue: 0,0:00:41.84,0:00:44.37,Default,,0000,0000,0000,,What all these vectors I&J? Dialogue: 0,0:00:44.97,0:00:47.90,Default,,0000,0000,0000,,Well, they have to be unit Dialogue: 0,0:00:47.90,0:00:51.84,Default,,0000,0000,0000,,vectors. A unit vector I under Dialogue: 0,0:00:51.84,0:00:54.35,Default,,0000,0000,0000,,unit vector. J. Dialogue: 0,0:00:55.57,0:00:58.79,Default,,0000,0000,0000,,In order to make sure that\Nwe do know that they are Dialogue: 0,0:00:58.79,0:01:01.47,Default,,0000,0000,0000,,unit vectors, we can put\Nlittle hat on the top. Dialogue: 0,0:01:03.28,0:01:05.90,Default,,0000,0000,0000,,So if we have a point P. Dialogue: 0,0:01:07.02,0:01:13.09,Default,,0000,0000,0000,,And let's say the coordinates of\Nthat point are three, 4, then Dialogue: 0,0:01:13.09,0:01:18.66,Default,,0000,0000,0000,,the position vector of P which\Nremember is that line segment Dialogue: 0,0:01:18.66,0:01:22.20,Default,,0000,0000,0000,,joining oh to pee is 3 I. Dialogue: 0,0:01:23.41,0:01:26.50,Default,,0000,0000,0000,,Plus four jazz. Dialogue: 0,0:01:28.32,0:01:32.50,Default,,0000,0000,0000,,Notice the crucial difference.\NThat's a set of coordinates Dialogue: 0,0:01:32.50,0:01:37.16,Default,,0000,0000,0000,,which refers to the point\Nthat's the vector which refers Dialogue: 0,0:01:37.16,0:01:41.80,Default,,0000,0000,0000,,to the position vector. So\Npoint and position vector are Dialogue: 0,0:01:41.80,0:01:43.66,Default,,0000,0000,0000,,not the same thing. Dialogue: 0,0:01:44.69,0:01:50.94,Default,,0000,0000,0000,,We can write this as\Na column vector 34. Dialogue: 0,0:01:51.91,0:01:56.78,Default,,0000,0000,0000,,And sometimes. This is\Nused sometimes that one Dialogue: 0,0:01:56.78,0:01:58.68,Default,,0000,0000,0000,,is used, just depends. Dialogue: 0,0:01:59.75,0:02:03.32,Default,,0000,0000,0000,,What about moving then into 3 Dialogue: 0,0:02:03.32,0:02:08.14,Default,,0000,0000,0000,,dimensions? We've got XY\Nand of course the tradition Dialogue: 0,0:02:08.14,0:02:09.96,Default,,0000,0000,0000,,is to use zed. Dialogue: 0,0:02:14.17,0:02:20.23,Default,,0000,0000,0000,,So let's have a look. Let's draw\Nin our three axes. Dialogue: 0,0:02:21.63,0:02:25.07,Default,,0000,0000,0000,,So then we've got XY. Dialogue: 0,0:02:25.78,0:02:27.65,Default,,0000,0000,0000,,And zed. Dialogue: 0,0:02:28.80,0:02:33.65,Default,,0000,0000,0000,,I'll always write zed with a bar\Nthrough it that so it doesn't Dialogue: 0,0:02:33.65,0:02:38.12,Default,,0000,0000,0000,,get mixed up with two. I don't\Nwant the letters Ed being Dialogue: 0,0:02:38.12,0:02:39.99,Default,,0000,0000,0000,,confused with the number 2. Dialogue: 0,0:02:40.95,0:02:47.16,Default,,0000,0000,0000,,So I've got these three axes or\Nat right angles to each other Dialogue: 0,0:02:47.16,0:02:52.90,Default,,0000,0000,0000,,and meeting at this origin. Oh,\Nand of course I'm going to Dialogue: 0,0:02:52.90,0:02:57.68,Default,,0000,0000,0000,,describe any point P by three\Ncoordinates XY and Z. Dialogue: 0,0:02:58.50,0:03:02.93,Default,,0000,0000,0000,,Now when I drew up this set of\Naxes, I indicated them. Dialogue: 0,0:03:03.49,0:03:04.39,Default,,0000,0000,0000,,Quite easily. Dialogue: 0,0:03:06.02,0:03:09.20,Default,,0000,0000,0000,,I could of course Interchange Dialogue: 0,0:03:09.20,0:03:12.55,Default,,0000,0000,0000,,X&Y. I might choose to Dialogue: 0,0:03:12.55,0:03:18.47,Default,,0000,0000,0000,,interchange Y&Z. But this is the\Nstandard way. Why is it the Dialogue: 0,0:03:18.47,0:03:23.44,Default,,0000,0000,0000,,standard way? What is it about\Nthis that makes it the standard Dialogue: 0,0:03:23.44,0:03:27.58,Default,,0000,0000,0000,,way? It's standard because it's\Nwhat we call a right? Dialogue: 0,0:03:29.39,0:03:30.52,Default,,0000,0000,0000,,Hand. Dialogue: 0,0:03:32.00,0:03:32.67,Default,,0000,0000,0000,,System. Dialogue: 0,0:03:35.33,0:03:41.99,Default,,0000,0000,0000,,Now. How can we describe\Nworkout? What is a right hand Dialogue: 0,0:03:41.99,0:03:47.52,Default,,0000,0000,0000,,system? Take your right hand and\Nhold it like this. Dialogue: 0,0:03:49.78,0:03:54.61,Default,,0000,0000,0000,,Middle finger. Full finger and\Nthumb at right angles. Dialogue: 0,0:03:56.66,0:03:57.94,Default,,0000,0000,0000,,This is the X axis. Dialogue: 0,0:03:58.97,0:04:03.03,Default,,0000,0000,0000,,The middle finger. This is the Y\Naxis, the thumb. Dialogue: 0,0:04:04.21,0:04:10.07,Default,,0000,0000,0000,,Now rotate as though we were\Nturning in a right handed screw. Dialogue: 0,0:04:10.71,0:04:12.60,Default,,0000,0000,0000,,And we rotate like that. Dialogue: 0,0:04:14.53,0:04:19.52,Default,,0000,0000,0000,,And so the direction in which\Nwe're moving this direction Dialogue: 0,0:04:19.52,0:04:22.51,Default,,0000,0000,0000,,becomes zed axis. So we rotate Dialogue: 0,0:04:22.51,0:04:25.18,Default,,0000,0000,0000,,from X. Why? Dialogue: 0,0:04:26.14,0:04:31.54,Default,,0000,0000,0000,,And we move in the direction of\Nthe Z axis. So right hander Dialogue: 0,0:04:31.54,0:04:35.68,Default,,0000,0000,0000,,rotation as those screwing in a\Nscrew right? Handedly notice Dialogue: 0,0:04:35.68,0:04:41.50,Default,,0000,0000,0000,,that it works whatever access we\Nchoose. So if we take this to be Dialogue: 0,0:04:41.50,0:04:48.14,Default,,0000,0000,0000,,Y again, the thumb and we take\Nthis to be zed then if we make a Dialogue: 0,0:04:48.14,0:04:53.12,Default,,0000,0000,0000,,right handed rotation from why\Nroute to zed, we will move along Dialogue: 0,0:04:53.12,0:04:56.02,Default,,0000,0000,0000,,the X axis. So let's do that. Dialogue: 0,0:04:56.35,0:05:02.49,Default,,0000,0000,0000,,You can see that as we rotate\Nit, we are moving right handedly Dialogue: 0,0:05:02.49,0:05:09.09,Default,,0000,0000,0000,,along the X axis and you can try\Nthe same for yourself in terms Dialogue: 0,0:05:09.09,0:05:15.23,Default,,0000,0000,0000,,of rotating from X to zed and\Nmoving along the Y axis. So Dialogue: 0,0:05:15.23,0:05:21.37,Default,,0000,0000,0000,,that's our right handed system.\NSo let's have a look at that in Dialogue: 0,0:05:21.37,0:05:27.03,Default,,0000,0000,0000,,terms of having a point P that's\Ngot its three coordinates XY. Dialogue: 0,0:05:27.12,0:05:27.79,Default,,0000,0000,0000,,And said Dialogue: 0,0:05:39.11,0:05:39.92,Default,,0000,0000,0000,,X. Dialogue: 0,0:05:41.86,0:05:42.97,Default,,0000,0000,0000,,And why? Dialogue: 0,0:05:45.44,0:05:48.35,Default,,0000,0000,0000,,And said now origin, oh. Dialogue: 0,0:05:49.21,0:05:55.76,Default,,0000,0000,0000,,Will take a point P anywhere\Nthere in space. What we're Dialogue: 0,0:05:55.76,0:06:02.30,Default,,0000,0000,0000,,interested in is this point P.\NIt's got coordinates, XY and Dialogue: 0,0:06:02.30,0:06:10.07,Default,,0000,0000,0000,,zed. And its position\Nvector is that line segment Dialogue: 0,0:06:10.07,0:06:13.47,Default,,0000,0000,0000,,OP. And so we can write down, Dialogue: 0,0:06:13.47,0:06:18.94,Default,,0000,0000,0000,,Oh, P. Bar is\Nequal to XI. Dialogue: 0,0:06:20.54,0:06:27.12,Default,,0000,0000,0000,,Plus YJ.\NPlus, Zed and the unit vector Dialogue: 0,0:06:27.12,0:06:33.28,Default,,0000,0000,0000,,that is in the direction of the\NZed Axis is taken to be K. Dialogue: 0,0:06:34.38,0:06:38.60,Default,,0000,0000,0000,,So again, notice the\Ndifference. These are the Dialogue: 0,0:06:38.60,0:06:42.81,Default,,0000,0000,0000,,coordinates XY, zed. This is\Nthe position vector Dialogue: 0,0:06:42.81,0:06:45.97,Default,,0000,0000,0000,,coordinates and position\Nvector are different. Dialogue: 0,0:06:46.99,0:06:50.18,Default,,0000,0000,0000,,Coordinates signify\Nappoint, position vector Dialogue: 0,0:06:50.18,0:06:55.27,Default,,0000,0000,0000,,signifies a line segment.\NWe sometimes write again Dialogue: 0,0:06:55.27,0:07:00.37,Default,,0000,0000,0000,,as we did with two\Ndimensions. We sometimes Dialogue: 0,0:07:00.37,0:07:05.46,Default,,0000,0000,0000,,write this as a column\Nvector XY zed. Dialogue: 0,0:07:07.07,0:07:12.15,Default,,0000,0000,0000,,Now there are various things we\Nwould like to know and certain Dialogue: 0,0:07:12.15,0:07:16.38,Default,,0000,0000,0000,,notation that we want to\Nintroduce for start. What's the Dialogue: 0,0:07:16.38,0:07:21.03,Default,,0000,0000,0000,,magnitude of Opie bar? What's\Nthe length of OP? Well, let's Dialogue: 0,0:07:21.03,0:07:25.68,Default,,0000,0000,0000,,drop a perpendicular down into\Nthe XY plane there and then. Dialogue: 0,0:07:25.68,0:07:27.37,Default,,0000,0000,0000,,Let's join this up. Dialogue: 0,0:07:30.15,0:07:35.41,Default,,0000,0000,0000,,The axes there and\Nacross there. Dialogue: 0,0:07:36.63,0:07:40.71,Default,,0000,0000,0000,,Now let's just think what this\Nmeans this length here. Dialogue: 0,0:07:41.34,0:07:45.68,Default,,0000,0000,0000,,Is the distance of the point\Nabove the XY plane, so it must Dialogue: 0,0:07:45.68,0:07:47.02,Default,,0000,0000,0000,,be of length zed. Dialogue: 0,0:07:48.82,0:07:54.07,Default,,0000,0000,0000,,This length, here and here is\Nthe same length. It's the Dialogue: 0,0:07:54.07,0:07:58.84,Default,,0000,0000,0000,,distance along the X coordinate,\Nso that must be X. Dialogue: 0,0:07:59.79,0:08:06.22,Default,,0000,0000,0000,,And that's also X. Similarly,\NThis is why and so that must be Dialogue: 0,0:08:06.22,0:08:07.71,Default,,0000,0000,0000,,why as well. Dialogue: 0,0:08:08.62,0:08:11.79,Default,,0000,0000,0000,,So if we join up from here out Dialogue: 0,0:08:11.79,0:08:15.85,Default,,0000,0000,0000,,to here. What we have here is a\Nright angle triangle, and of Dialogue: 0,0:08:15.85,0:08:17.23,Default,,0000,0000,0000,,course we've got a right angle Dialogue: 0,0:08:17.23,0:08:21.66,Default,,0000,0000,0000,,triangle here as well. So this\Nlength here. There's are right Dialogue: 0,0:08:21.66,0:08:25.34,Default,,0000,0000,0000,,angle this length using\NPythagoras must be the square Dialogue: 0,0:08:25.34,0:08:29.84,Default,,0000,0000,0000,,root of X squared plus Y\Nsquared, and so because we've Dialogue: 0,0:08:29.84,0:08:34.75,Default,,0000,0000,0000,,got a right angle here, if we\Nuse Pythagoras in this triangle Dialogue: 0,0:08:34.75,0:08:40.07,Default,,0000,0000,0000,,then we end up with the fact\Nthat opie, the modulus of Opie Dialogue: 0,0:08:40.07,0:08:46.20,Default,,0000,0000,0000,,Bar is the square root of. We've\Ngot to square that and add it to Dialogue: 0,0:08:46.20,0:08:50.29,Default,,0000,0000,0000,,the square of that. So that's\Njust X squared plus. Dialogue: 0,0:08:50.38,0:08:53.22,Default,,0000,0000,0000,,Y squared plus Zed Square. Dialogue: 0,0:08:56.59,0:09:00.36,Default,,0000,0000,0000,,Now I'm going to draw this\Ndiagram again, but I'm going to Dialogue: 0,0:09:00.36,0:09:04.13,Default,,0000,0000,0000,,try and miss out some of the\Nextra lines that we've added. Dialogue: 0,0:09:12.83,0:09:16.18,Default,,0000,0000,0000,,So XY. Dialogue: 0,0:09:17.77,0:09:18.56,Default,,0000,0000,0000,,Zedd. Dialogue: 0,0:09:19.77,0:09:25.87,Default,,0000,0000,0000,,We'll take our point P with\Nposition vector OP bar. Dialogue: 0,0:09:27.83,0:09:28.55,Default,,0000,0000,0000,,Again. Dialogue: 0,0:09:29.80,0:09:34.80,Default,,0000,0000,0000,,Drop that perpendicular down on\Nto the XY plane. Dialogue: 0,0:09:36.37,0:09:38.42,Default,,0000,0000,0000,,Draw this in across here. Dialogue: 0,0:09:39.47,0:09:41.70,Default,,0000,0000,0000,,And that in there. Dialogue: 0,0:09:43.09,0:09:50.14,Default,,0000,0000,0000,,Now. This line OP\Nmakes an angle with this Dialogue: 0,0:09:50.14,0:09:51.57,Default,,0000,0000,0000,,axis here. Dialogue: 0,0:09:54.09,0:09:56.36,Default,,0000,0000,0000,,It makes an angle Alpha. Dialogue: 0,0:09:57.49,0:09:59.40,Default,,0000,0000,0000,,And if I draw it out so that we Dialogue: 0,0:09:59.40,0:10:02.45,Default,,0000,0000,0000,,can see it. Let me call this a. Dialogue: 0,0:10:03.33,0:10:06.92,Default,,0000,0000,0000,,If we draw out the triangle\Nso that we can actually see Dialogue: 0,0:10:06.92,0:10:09.31,Default,,0000,0000,0000,,what we've got, then we've\Ngot the line. Dialogue: 0,0:10:10.50,0:10:13.14,Default,,0000,0000,0000,,From O to a. Dialogue: 0,0:10:14.96,0:10:21.46,Default,,0000,0000,0000,,There. Oh, to A and we've got\Nthis line going out here from A Dialogue: 0,0:10:21.46,0:10:26.64,Default,,0000,0000,0000,,to pee and that's going to be at\Nright angles there like that. Dialogue: 0,0:10:27.46,0:10:33.45,Default,,0000,0000,0000,,And so if we now join P2O, we\Ncan see the angle here, Alpha. Dialogue: 0,0:10:34.96,0:10:40.06,Default,,0000,0000,0000,,Now we know the length of this\Nline. We know that it is the Dialogue: 0,0:10:40.06,0:10:44.79,Default,,0000,0000,0000,,square root of X squared plus Y\Nsquared plus zed squared and we Dialogue: 0,0:10:44.79,0:10:48.06,Default,,0000,0000,0000,,also know the length of this\Nline, it's X. Dialogue: 0,0:10:48.98,0:10:55.87,Default,,0000,0000,0000,,And that is a right angle, and\Nso therefore we can write down Dialogue: 0,0:10:55.87,0:11:02.76,Default,,0000,0000,0000,,cause of Alpha is equal to X\Nover square root of X squared Dialogue: 0,0:11:02.76,0:11:05.94,Default,,0000,0000,0000,,plus Y squared plus zed squared. Dialogue: 0,0:11:07.50,0:11:14.69,Default,,0000,0000,0000,,Why have we chosen this? Well,\Ncause Alpha is what is known Dialogue: 0,0:11:14.69,0:11:16.48,Default,,0000,0000,0000,,as a direction. Dialogue: 0,0:11:18.79,0:11:19.59,Default,,0000,0000,0000,,Cosine Dialogue: 0,0:11:22.40,0:11:29.28,Default,,0000,0000,0000,,be cause. It is the cosine of\Nan angle that in some way helps Dialogue: 0,0:11:29.28,0:11:32.05,Default,,0000,0000,0000,,to specify the direction of P. Dialogue: 0,0:11:33.03,0:11:38.29,Default,,0000,0000,0000,,An Alpha is the angle that Opie\Nmakes with the X axis. So of Dialogue: 0,0:11:38.29,0:11:44.31,Default,,0000,0000,0000,,course what we can do for the X\Naxis we can do for the Y axis Dialogue: 0,0:11:44.31,0:11:46.19,Default,,0000,0000,0000,,and for the Z Axis. Dialogue: 0,0:11:48.65,0:11:55.26,Default,,0000,0000,0000,,So we have calls Alpha which\Nwill be X over the square Dialogue: 0,0:11:55.26,0:12:00.77,Default,,0000,0000,0000,,root of X squared plus Y\Nsquared plus zed squared. Dialogue: 0,0:12:02.60,0:12:08.95,Default,,0000,0000,0000,,Kohl's beta which will be\Nthe angle that Opie makes with Dialogue: 0,0:12:08.95,0:12:15.89,Default,,0000,0000,0000,,the Y axis, and so it will be\Nwhy over the square root of X Dialogue: 0,0:12:15.89,0:12:20.52,Default,,0000,0000,0000,,squared plus Y squared plus zed\Nsquared and cause gamma. Dialogue: 0,0:12:21.38,0:12:27.23,Default,,0000,0000,0000,,Gamma is the angle that Opie\Nmakes with the Z Axis, and so it Dialogue: 0,0:12:27.23,0:12:32.67,Default,,0000,0000,0000,,will be zed over the square root\Nof X squared plus Y squared Dialogue: 0,0:12:32.67,0:12:33.92,Default,,0000,0000,0000,,close zed square. Dialogue: 0,0:12:35.47,0:12:40.09,Default,,0000,0000,0000,,So these are our direction\Ncosines. These are expressions Dialogue: 0,0:12:40.09,0:12:45.22,Default,,0000,0000,0000,,for being able to calculate\Nthem, but there is something Dialogue: 0,0:12:45.22,0:12:51.37,Default,,0000,0000,0000,,that we can notice about them.\NWhat happens if we square them Dialogue: 0,0:12:51.37,0:12:58.04,Default,,0000,0000,0000,,and add them? So what do we\Nget if we take 'cause squared Dialogue: 0,0:12:58.04,0:13:02.66,Default,,0000,0000,0000,,Alpha plus cause squared beta\Nplus cause squared gamma? Dialogue: 0,0:13:04.08,0:13:06.27,Default,,0000,0000,0000,,So let's just calculate\Nthis expression. Dialogue: 0,0:13:09.34,0:13:16.07,Default,,0000,0000,0000,,Kohl's squared Alpha is going\Nto be X squared over Dialogue: 0,0:13:16.07,0:13:21.45,Default,,0000,0000,0000,,X squared plus Y squared\Nplus said squared. Dialogue: 0,0:13:23.62,0:13:31.56,Default,,0000,0000,0000,,Call squared beta is going to\Nbe Y squared over X squared Dialogue: 0,0:13:31.56,0:13:35.54,Default,,0000,0000,0000,,plus Y squared plus zed squared. Dialogue: 0,0:13:36.36,0:13:42.30,Default,,0000,0000,0000,,And cost squared gamma is\Ngoing to be zed squared Dialogue: 0,0:13:42.30,0:13:47.65,Default,,0000,0000,0000,,over X squared plus Y\Nsquared plus zed squared. Dialogue: 0,0:13:49.82,0:13:55.79,Default,,0000,0000,0000,,Now we're looking at adding all\Nof these three expressions Dialogue: 0,0:13:55.79,0:14:01.76,Default,,0000,0000,0000,,together. Cost squared Alpha\Nplus cost squared beta plus cost Dialogue: 0,0:14:01.76,0:14:07.03,Default,,0000,0000,0000,,squared gamma. Well, they've all\Ngot exactly the same denominator Dialogue: 0,0:14:07.03,0:14:12.21,Default,,0000,0000,0000,,X squared plus Y squared plus\Nsaid squared, so we can just add Dialogue: 0,0:14:12.21,0:14:16.59,Default,,0000,0000,0000,,together X squared plus Y\Nsquared plus 10 squared in the Dialogue: 0,0:14:16.59,0:14:20.96,Default,,0000,0000,0000,,numerator. So that's X squared\Nplus Y squared zed squared all Dialogue: 0,0:14:20.96,0:14:25.34,Default,,0000,0000,0000,,over X squared plus Y squared\Nplus said squared. Of course, Dialogue: 0,0:14:25.34,0:14:26.54,Default,,0000,0000,0000,,that's just one. Dialogue: 0,0:14:27.63,0:14:32.16,Default,,0000,0000,0000,,So the squares of the direction\Ncosines added together give us Dialogue: 0,0:14:32.16,0:14:37.30,Default,,0000,0000,0000,,one. What possible use could\Nthat be to us? Well, one of Dialogue: 0,0:14:37.30,0:14:41.43,Default,,0000,0000,0000,,the things it does mean is\Nthat we have the vector, Dialogue: 0,0:14:41.43,0:14:42.56,Default,,0000,0000,0000,,let's say cause. Dialogue: 0,0:14:43.60,0:14:51.39,Default,,0000,0000,0000,,Alpha I plus\Ncause beta J Dialogue: 0,0:14:51.39,0:14:56.58,Default,,0000,0000,0000,,plus cause Gamma\NK. Dialogue: 0,0:14:58.01,0:15:03.06,Default,,0000,0000,0000,,That vector is a unit vector.\NIt's a unit vector because if Dialogue: 0,0:15:03.06,0:15:06.85,Default,,0000,0000,0000,,we calculate its magnitude\Nthat's cost squared Alpha plus Dialogue: 0,0:15:06.85,0:15:11.48,Default,,0000,0000,0000,,cost squared beta plus cost\Nsquared gamma is equal to 1. Dialogue: 0,0:15:11.48,0:15:16.53,Default,,0000,0000,0000,,Take the square root. That's\None. So this is a unit vector. Dialogue: 0,0:15:16.53,0:15:21.59,Default,,0000,0000,0000,,Further, this is X over X\Nsquared plus Y squared plus Z Dialogue: 0,0:15:21.59,0:15:26.22,Default,,0000,0000,0000,,squared Y over X squared plus\NY squared plus said squared. Dialogue: 0,0:15:27.47,0:15:33.59,Default,,0000,0000,0000,,And zed over X squared plus Y\Nsquared plus said Square and so Dialogue: 0,0:15:33.59,0:15:38.77,Default,,0000,0000,0000,,it's in the same direction as\Nour original OP. Our original Dialogue: 0,0:15:38.77,0:15:40.66,Default,,0000,0000,0000,,position vector opi bar. Dialogue: 0,0:15:41.44,0:15:46.72,Default,,0000,0000,0000,,And that means that this is a\Nunit vector in the direction of Dialogue: 0,0:15:46.72,0:15:52.40,Default,,0000,0000,0000,,OP bar and that may prove to be\Nquite useful later on when we Dialogue: 0,0:15:52.40,0:15:56.87,Default,,0000,0000,0000,,want to look at unit vectors in\Nparticular directions. For now, Dialogue: 0,0:15:56.87,0:16:01.74,Default,,0000,0000,0000,,let's just have a look at doing\Na little bit of calculation. Dialogue: 0,0:16:05.62,0:16:07.30,Default,,0000,0000,0000,,Let's say we've got a point. Dialogue: 0,0:16:08.39,0:16:13.73,Default,,0000,0000,0000,,That has\Ncoordinates 102. Dialogue: 0,0:16:14.82,0:16:21.43,Default,,0000,0000,0000,,Under point that has\Ncoordinates 2 - 1. Dialogue: 0,0:16:22.75,0:16:23.27,Default,,0000,0000,0000,,4. Dialogue: 0,0:16:24.73,0:16:30.90,Default,,0000,0000,0000,,The question that we might ask\Nis if we form the vector AB. Dialogue: 0,0:16:31.62,0:16:33.55,Default,,0000,0000,0000,,What's the magnitude of a bee? Dialogue: 0,0:16:34.20,0:16:36.25,Default,,0000,0000,0000,,And what are its direction Dialogue: 0,0:16:36.25,0:16:40.98,Default,,0000,0000,0000,,cosines? We just have a\Nlook at this. Let's Dialogue: 0,0:16:40.98,0:16:43.18,Default,,0000,0000,0000,,remember that, oh, a bar. Dialogue: 0,0:16:44.20,0:16:44.83,Default,,0000,0000,0000,,Is. Dialogue: 0,0:16:46.44,0:16:47.20,Default,,0000,0000,0000,,I. Dialogue: 0,0:16:48.26,0:16:49.57,Default,,0000,0000,0000,,No JS. Dialogue: 0,0:16:50.94,0:16:53.53,Default,,0000,0000,0000,,And two K's. Dialogue: 0,0:16:54.27,0:16:56.42,Default,,0000,0000,0000,,That OB bar. Dialogue: 0,0:16:57.38,0:17:00.07,Default,,0000,0000,0000,,Will be. Two I. Dialogue: 0,0:17:01.86,0:17:07.77,Default,,0000,0000,0000,,Minus one\NJ plus 4K. Dialogue: 0,0:17:09.49,0:17:15.81,Default,,0000,0000,0000,,We want to know what's the\Nmagnitude of the vector AB bar. Dialogue: 0,0:17:17.05,0:17:22.13,Default,,0000,0000,0000,,Just draw quick picture just to\Nremind ourselves of how to get Dialogue: 0,0:17:22.13,0:17:26.36,Default,,0000,0000,0000,,there. There's A and its\Nposition vector with respect to. Dialogue: 0,0:17:26.36,0:17:31.43,Default,,0000,0000,0000,,Oh there's B with its position\Nvector with respect to. If we're Dialogue: 0,0:17:31.43,0:17:37.35,Default,,0000,0000,0000,,wanting a baby that's from there\Nto there and so we can see that Dialogue: 0,0:17:37.35,0:17:42.85,Default,,0000,0000,0000,,by going from A to B, we can go\Nround AO plus OB. Dialogue: 0,0:17:44.34,0:17:50.99,Default,,0000,0000,0000,,And so therefore, that is OB bar\Nminus Oh, a bar. So that's what Dialogue: 0,0:17:50.99,0:17:58.12,Default,,0000,0000,0000,,we need to do here. A bar must\Nbe OB bar minus oh, a bar. Dialogue: 0,0:17:59.13,0:18:02.80,Default,,0000,0000,0000,,And all we do to do the\Nsubtraction is what you would Dialogue: 0,0:18:02.80,0:18:05.56,Default,,0000,0000,0000,,do naturally, which is to\Nsubtract the respective bits Dialogue: 0,0:18:05.56,0:18:09.23,Default,,0000,0000,0000,,so it's two I take away I.\NThat's just an eye bar. Dialogue: 0,0:18:11.17,0:18:17.42,Default,,0000,0000,0000,,Minus J takeaway no\NJS, so that's minus J Dialogue: 0,0:18:17.42,0:18:22.98,Default,,0000,0000,0000,,Bar and 4K takeaway\N2K. That's plus 2K. Dialogue: 0,0:18:24.19,0:18:30.65,Default,,0000,0000,0000,,So now we have our vector AB\Nbar. We can calculate its Dialogue: 0,0:18:30.65,0:18:37.64,Default,,0000,0000,0000,,magnitude AB modulus of a bar\Nthat's just a be the length from Dialogue: 0,0:18:37.64,0:18:45.17,Default,,0000,0000,0000,,A to B, and that's the square\Nroot of 1 squared plus minus one Dialogue: 0,0:18:45.17,0:18:51.09,Default,,0000,0000,0000,,squared +2 squared altogether.\NThat's 1 + 1 + 4 square Dialogue: 0,0:18:51.09,0:18:54.32,Default,,0000,0000,0000,,root of 6, and the direction Dialogue: 0,0:18:54.32,0:18:57.48,Default,,0000,0000,0000,,cosines. Our cause Alpha. Dialogue: 0,0:18:59.71,0:19:04.37,Default,,0000,0000,0000,,That's The X coordinate\Nover the modulus, so that's Dialogue: 0,0:19:04.37,0:19:06.44,Default,,0000,0000,0000,,one over Route 6. Dialogue: 0,0:19:08.63,0:19:14.81,Default,,0000,0000,0000,,Kohl's beta that's minus\None over Route 6. the Y Dialogue: 0,0:19:14.81,0:19:19.14,Default,,0000,0000,0000,,coordinate over the\Nmodulus and cause gamma. Dialogue: 0,0:19:20.68,0:19:25.54,Default,,0000,0000,0000,,The zed coordinate\Nover the modulus. Dialogue: 0,0:19:27.27,0:19:31.47,Default,,0000,0000,0000,,Now this is a fairly standard\Ncalculation. The sort of Dialogue: 0,0:19:31.47,0:19:33.57,Default,,0000,0000,0000,,calculation that it will be Dialogue: 0,0:19:33.57,0:19:38.09,Default,,0000,0000,0000,,expected. You'll be able to do\Nand simply be able to work your Dialogue: 0,0:19:38.09,0:19:41.34,Default,,0000,0000,0000,,way through it very quickly.\NVery, very easily, so you have Dialogue: 0,0:19:41.34,0:19:45.46,Default,,0000,0000,0000,,to be able to practice some of\Nthese. You have to be able to Dialogue: 0,0:19:45.46,0:19:48.71,Default,,0000,0000,0000,,work with it very rapidly, very,\Nvery easily, but always keep Dialogue: 0,0:19:48.71,0:19:49.89,Default,,0000,0000,0000,,this diagram in mind. Dialogue: 0,0:19:50.71,0:19:57.75,Default,,0000,0000,0000,,That to get from A to B to form\Nthe vector AB bar, you go a Dialogue: 0,0:19:57.75,0:19:59.95,Default,,0000,0000,0000,,obarr plus Obiba and so. Dialogue: 0,0:20:00.18,0:20:07.59,Default,,0000,0000,0000,,It's the result, so to form a B\Nit's Obi bar, take away OA bar.