1 00:00:02,260 --> 00:00:06,914 The natural way to describe the position of any point is to use 2 00:00:06,914 --> 00:00:10,858 Cartesian coordinates. In two dimensions it's quite easy. 3 00:00:11,580 --> 00:00:17,851 We just have. Picture like this and so we have an X 4 00:00:17,851 --> 00:00:19,807 axis and Y Axis. 5 00:00:22,130 --> 00:00:27,460 Origin oh where they cross and if we want to have vectors in 6 00:00:27,460 --> 00:00:31,970 that arrangement, what we would have is a vector I associated 7 00:00:31,970 --> 00:00:37,300 with the X axis and a vector Jay associated with the Y axis. 8 00:00:41,840 --> 00:00:44,370 What all these vectors I&J? 9 00:00:44,970 --> 00:00:47,898 Well, they have to be unit 10 00:00:47,898 --> 00:00:51,840 vectors. A unit vector I under 11 00:00:51,840 --> 00:00:54,350 unit vector. J. 12 00:00:55,570 --> 00:00:58,786 In order to make sure that we do know that they are 13 00:00:58,786 --> 00:01:01,466 unit vectors, we can put little hat on the top. 14 00:01:03,280 --> 00:01:05,898 So if we have a point P. 15 00:01:07,020 --> 00:01:13,092 And let's say the coordinates of that point are three, 4, then 16 00:01:13,092 --> 00:01:18,658 the position vector of P which remember is that line segment 17 00:01:18,658 --> 00:01:22,200 joining oh to pee is 3 I. 18 00:01:23,410 --> 00:01:26,500 Plus four jazz. 19 00:01:28,320 --> 00:01:32,505 Notice the crucial difference. That's a set of coordinates 20 00:01:32,505 --> 00:01:37,155 which refers to the point that's the vector which refers 21 00:01:37,155 --> 00:01:41,805 to the position vector. So point and position vector are 22 00:01:41,805 --> 00:01:43,665 not the same thing. 23 00:01:44,690 --> 00:01:50,945 We can write this as a column vector 34. 24 00:01:51,910 --> 00:01:56,776 And sometimes. This is used sometimes that one 25 00:01:56,776 --> 00:01:58,680 is used, just depends. 26 00:01:59,750 --> 00:02:03,320 What about moving then into 3 27 00:02:03,320 --> 00:02:08,136 dimensions? We've got XY and of course the tradition 28 00:02:08,136 --> 00:02:09,964 is to use zed. 29 00:02:14,170 --> 00:02:20,231 So let's have a look. Let's draw in our three axes. 30 00:02:21,630 --> 00:02:25,070 So then we've got XY. 31 00:02:25,780 --> 00:02:27,650 And zed. 32 00:02:28,800 --> 00:02:33,649 I'll always write zed with a bar through it that so it doesn't 33 00:02:33,649 --> 00:02:38,125 get mixed up with two. I don't want the letters Ed being 34 00:02:38,125 --> 00:02:39,990 confused with the number 2. 35 00:02:40,950 --> 00:02:47,164 So I've got these three axes or at right angles to each other 36 00:02:47,164 --> 00:02:52,900 and meeting at this origin. Oh, and of course I'm going to 37 00:02:52,900 --> 00:02:57,680 describe any point P by three coordinates XY and Z. 38 00:02:58,500 --> 00:03:02,928 Now when I drew up this set of axes, I indicated them. 39 00:03:03,490 --> 00:03:04,390 Quite easily. 40 00:03:06,020 --> 00:03:09,195 I could of course Interchange 41 00:03:09,195 --> 00:03:12,550 X&Y. I might choose to 42 00:03:12,550 --> 00:03:18,470 interchange Y&Z. But this is the standard way. Why is it the 43 00:03:18,470 --> 00:03:23,438 standard way? What is it about this that makes it the standard 44 00:03:23,438 --> 00:03:27,578 way? It's standard because it's what we call a right? 45 00:03:29,390 --> 00:03:30,520 Hand. 46 00:03:32,000 --> 00:03:32,670 System. 47 00:03:35,330 --> 00:03:41,990 Now. How can we describe workout? What is a right hand 48 00:03:41,990 --> 00:03:47,520 system? Take your right hand and hold it like this. 49 00:03:49,780 --> 00:03:54,608 Middle finger. Full finger and thumb at right angles. 50 00:03:56,660 --> 00:03:57,940 This is the X axis. 51 00:03:58,970 --> 00:04:03,030 The middle finger. This is the Y axis, the thumb. 52 00:04:04,210 --> 00:04:10,066 Now rotate as though we were turning in a right handed screw. 53 00:04:10,710 --> 00:04:12,600 And we rotate like that. 54 00:04:14,530 --> 00:04:19,520 And so the direction in which we're moving this direction 55 00:04:19,520 --> 00:04:22,514 becomes zed axis. So we rotate 56 00:04:22,514 --> 00:04:25,180 from X. Why? 57 00:04:26,140 --> 00:04:31,535 And we move in the direction of the Z axis. So right hander 58 00:04:31,535 --> 00:04:35,685 rotation as those screwing in a screw right? Handedly notice 59 00:04:35,685 --> 00:04:41,495 that it works whatever access we choose. So if we take this to be 60 00:04:41,495 --> 00:04:48,135 Y again, the thumb and we take this to be zed then if we make a 61 00:04:48,135 --> 00:04:53,115 right handed rotation from why route to zed, we will move along 62 00:04:53,115 --> 00:04:56,020 the X axis. So let's do that. 63 00:04:56,350 --> 00:05:02,486 You can see that as we rotate it, we are moving right handedly 64 00:05:02,486 --> 00:05:09,094 along the X axis and you can try the same for yourself in terms 65 00:05:09,094 --> 00:05:15,230 of rotating from X to zed and moving along the Y axis. So 66 00:05:15,230 --> 00:05:21,366 that's our right handed system. So let's have a look at that in 67 00:05:21,366 --> 00:05:27,030 terms of having a point P that's got its three coordinates XY. 68 00:05:27,120 --> 00:05:27,790 And said 69 00:05:39,110 --> 00:05:39,920 X. 70 00:05:41,860 --> 00:05:42,970 And why? 71 00:05:45,440 --> 00:05:48,350 And said now origin, oh. 72 00:05:49,210 --> 00:05:55,755 Will take a point P anywhere there in space. What we're 73 00:05:55,755 --> 00:06:02,300 interested in is this point P. It's got coordinates, XY and 74 00:06:02,300 --> 00:06:10,074 zed. And its position vector is that line segment 75 00:06:10,074 --> 00:06:13,468 OP. And so we can write down, 76 00:06:13,468 --> 00:06:18,940 Oh, P. Bar is equal to XI. 77 00:06:20,540 --> 00:06:27,120 Plus YJ. Plus, Zed and the unit vector 78 00:06:27,120 --> 00:06:33,280 that is in the direction of the Zed Axis is taken to be K. 79 00:06:34,380 --> 00:06:38,596 So again, notice the difference. These are the 80 00:06:38,596 --> 00:06:42,812 coordinates XY, zed. This is the position vector 81 00:06:42,812 --> 00:06:45,974 coordinates and position vector are different. 82 00:06:46,990 --> 00:06:50,175 Coordinates signify appoint, position vector 83 00:06:50,175 --> 00:06:55,271 signifies a line segment. We sometimes write again 84 00:06:55,271 --> 00:07:00,367 as we did with two dimensions. We sometimes 85 00:07:00,367 --> 00:07:05,463 write this as a column vector XY zed. 86 00:07:07,070 --> 00:07:12,146 Now there are various things we would like to know and certain 87 00:07:12,146 --> 00:07:16,376 notation that we want to introduce for start. What's the 88 00:07:16,376 --> 00:07:21,029 magnitude of Opie bar? What's the length of OP? Well, let's 89 00:07:21,029 --> 00:07:25,682 drop a perpendicular down into the XY plane there and then. 90 00:07:25,682 --> 00:07:27,374 Let's join this up. 91 00:07:30,150 --> 00:07:35,406 The axes there and across there. 92 00:07:36,630 --> 00:07:40,710 Now let's just think what this means this length here. 93 00:07:41,340 --> 00:07:45,682 Is the distance of the point above the XY plane, so it must 94 00:07:45,682 --> 00:07:47,018 be of length zed. 95 00:07:48,820 --> 00:07:54,067 This length, here and here is the same length. It's the 96 00:07:54,067 --> 00:07:58,837 distance along the X coordinate, so that must be X. 97 00:07:59,790 --> 00:08:06,225 And that's also X. Similarly, This is why and so that must be 98 00:08:06,225 --> 00:08:07,710 why as well. 99 00:08:08,620 --> 00:08:11,788 So if we join up from here out 100 00:08:11,788 --> 00:08:15,850 to here. What we have here is a right angle triangle, and of 101 00:08:15,850 --> 00:08:17,230 course we've got a right angle 102 00:08:17,230 --> 00:08:21,663 triangle here as well. So this length here. There's are right 103 00:08:21,663 --> 00:08:25,344 angle this length using Pythagoras must be the square 104 00:08:25,344 --> 00:08:29,843 root of X squared plus Y squared, and so because we've 105 00:08:29,843 --> 00:08:34,751 got a right angle here, if we use Pythagoras in this triangle 106 00:08:34,751 --> 00:08:40,068 then we end up with the fact that opie, the modulus of Opie 107 00:08:40,068 --> 00:08:46,203 Bar is the square root of. We've got to square that and add it to 108 00:08:46,203 --> 00:08:50,293 the square of that. So that's just X squared plus. 109 00:08:50,380 --> 00:08:53,220 Y squared plus Zed Square. 110 00:08:56,590 --> 00:09:00,358 Now I'm going to draw this diagram again, but I'm going to 111 00:09:00,358 --> 00:09:04,126 try and miss out some of the extra lines that we've added. 112 00:09:12,830 --> 00:09:16,180 So XY. 113 00:09:17,770 --> 00:09:18,560 Zedd. 114 00:09:19,770 --> 00:09:25,870 We'll take our point P with position vector OP bar. 115 00:09:27,830 --> 00:09:28,550 Again. 116 00:09:29,800 --> 00:09:34,795 Drop that perpendicular down on to the XY plane. 117 00:09:36,370 --> 00:09:38,420 Draw this in across here. 118 00:09:39,470 --> 00:09:41,698 And that in there. 119 00:09:43,090 --> 00:09:50,142 Now. This line OP makes an angle with this 120 00:09:50,142 --> 00:09:51,570 axis here. 121 00:09:54,090 --> 00:09:56,360 It makes an angle Alpha. 122 00:09:57,490 --> 00:09:59,398 And if I draw it out so that we 123 00:09:59,398 --> 00:10:02,450 can see it. Let me call this a. 124 00:10:03,330 --> 00:10:06,918 If we draw out the triangle so that we can actually see 125 00:10:06,918 --> 00:10:09,310 what we've got, then we've got the line. 126 00:10:10,500 --> 00:10:13,140 From O to a. 127 00:10:14,960 --> 00:10:21,464 There. Oh, to A and we've got this line going out here from A 128 00:10:21,464 --> 00:10:26,638 to pee and that's going to be at right angles there like that. 129 00:10:27,460 --> 00:10:33,452 And so if we now join P2O, we can see the angle here, Alpha. 130 00:10:34,960 --> 00:10:40,056 Now we know the length of this line. We know that it is the 131 00:10:40,056 --> 00:10:44,788 square root of X squared plus Y squared plus zed squared and we 132 00:10:44,788 --> 00:10:48,064 also know the length of this line, it's X. 133 00:10:48,980 --> 00:10:55,870 And that is a right angle, and so therefore we can write down 134 00:10:55,870 --> 00:11:02,760 cause of Alpha is equal to X over square root of X squared 135 00:11:02,760 --> 00:11:05,940 plus Y squared plus zed squared. 136 00:11:07,500 --> 00:11:14,688 Why have we chosen this? Well, cause Alpha is what is known 137 00:11:14,688 --> 00:11:16,485 as a direction. 138 00:11:18,790 --> 00:11:19,590 Cosine 139 00:11:22,400 --> 00:11:29,282 be cause. It is the cosine of an angle that in some way helps 140 00:11:29,282 --> 00:11:32,048 to specify the direction of P. 141 00:11:33,030 --> 00:11:38,294 An Alpha is the angle that Opie makes with the X axis. So of 142 00:11:38,294 --> 00:11:44,310 course what we can do for the X axis we can do for the Y axis 143 00:11:44,310 --> 00:11:46,190 and for the Z Axis. 144 00:11:48,650 --> 00:11:55,262 So we have calls Alpha which will be X over the square 145 00:11:55,262 --> 00:12:00,772 root of X squared plus Y squared plus zed squared. 146 00:12:02,600 --> 00:12:08,947 Kohl's beta which will be the angle that Opie makes with 147 00:12:08,947 --> 00:12:15,892 the Y axis, and so it will be why over the square root of X 148 00:12:15,892 --> 00:12:20,522 squared plus Y squared plus zed squared and cause gamma. 149 00:12:21,380 --> 00:12:27,232 Gamma is the angle that Opie makes with the Z Axis, and so it 150 00:12:27,232 --> 00:12:32,666 will be zed over the square root of X squared plus Y squared 151 00:12:32,666 --> 00:12:33,920 close zed square. 152 00:12:35,470 --> 00:12:40,087 So these are our direction cosines. These are expressions 153 00:12:40,087 --> 00:12:45,217 for being able to calculate them, but there is something 154 00:12:45,217 --> 00:12:51,373 that we can notice about them. What happens if we square them 155 00:12:51,373 --> 00:12:58,042 and add them? So what do we get if we take 'cause squared 156 00:12:58,042 --> 00:13:02,659 Alpha plus cause squared beta plus cause squared gamma? 157 00:13:04,080 --> 00:13:06,270 So let's just calculate this expression. 158 00:13:09,340 --> 00:13:16,070 Kohl's squared Alpha is going to be X squared over 159 00:13:16,070 --> 00:13:21,454 X squared plus Y squared plus said squared. 160 00:13:23,620 --> 00:13:31,564 Call squared beta is going to be Y squared over X squared 161 00:13:31,564 --> 00:13:35,536 plus Y squared plus zed squared. 162 00:13:36,360 --> 00:13:42,300 And cost squared gamma is going to be zed squared 163 00:13:42,300 --> 00:13:47,646 over X squared plus Y squared plus zed squared. 164 00:13:49,820 --> 00:13:55,790 Now we're looking at adding all of these three expressions 165 00:13:55,790 --> 00:14:01,760 together. Cost squared Alpha plus cost squared beta plus cost 166 00:14:01,760 --> 00:14:07,034 squared gamma. Well, they've all got exactly the same denominator 167 00:14:07,034 --> 00:14:12,208 X squared plus Y squared plus said squared, so we can just add 168 00:14:12,208 --> 00:14:16,586 together X squared plus Y squared plus 10 squared in the 169 00:14:16,586 --> 00:14:20,964 numerator. So that's X squared plus Y squared zed squared all 170 00:14:20,964 --> 00:14:25,342 over X squared plus Y squared plus said squared. Of course, 171 00:14:25,342 --> 00:14:26,536 that's just one. 172 00:14:27,630 --> 00:14:32,162 So the squares of the direction cosines added together give us 173 00:14:32,162 --> 00:14:37,296 one. What possible use could that be to us? Well, one of 174 00:14:37,296 --> 00:14:41,432 the things it does mean is that we have the vector, 175 00:14:41,432 --> 00:14:42,560 let's say cause. 176 00:14:43,600 --> 00:14:51,388 Alpha I plus cause beta J 177 00:14:51,388 --> 00:14:56,580 plus cause Gamma K. 178 00:14:58,010 --> 00:15:03,062 That vector is a unit vector. It's a unit vector because if 179 00:15:03,062 --> 00:15:06,851 we calculate its magnitude that's cost squared Alpha plus 180 00:15:06,851 --> 00:15:11,482 cost squared beta plus cost squared gamma is equal to 1. 181 00:15:11,482 --> 00:15:16,534 Take the square root. That's one. So this is a unit vector. 182 00:15:16,534 --> 00:15:21,586 Further, this is X over X squared plus Y squared plus Z 183 00:15:21,586 --> 00:15:26,217 squared Y over X squared plus Y squared plus said squared. 184 00:15:27,470 --> 00:15:33,593 And zed over X squared plus Y squared plus said Square and so 185 00:15:33,593 --> 00:15:38,774 it's in the same direction as our original OP. Our original 186 00:15:38,774 --> 00:15:40,658 position vector opi bar. 187 00:15:41,440 --> 00:15:46,718 And that means that this is a unit vector in the direction of 188 00:15:46,718 --> 00:15:52,402 OP bar and that may prove to be quite useful later on when we 189 00:15:52,402 --> 00:15:56,868 want to look at unit vectors in particular directions. For now, 190 00:15:56,868 --> 00:16:01,740 let's just have a look at doing a little bit of calculation. 191 00:16:05,620 --> 00:16:07,300 Let's say we've got a point. 192 00:16:08,390 --> 00:16:13,730 That has coordinates 102. 193 00:16:14,820 --> 00:16:21,428 Under point that has coordinates 2 - 1. 194 00:16:22,750 --> 00:16:23,270 4. 195 00:16:24,730 --> 00:16:30,905 The question that we might ask is if we form the vector AB. 196 00:16:31,620 --> 00:16:33,546 What's the magnitude of a bee? 197 00:16:34,200 --> 00:16:36,250 And what are its direction 198 00:16:36,250 --> 00:16:40,982 cosines? We just have a look at this. Let's 199 00:16:40,982 --> 00:16:43,177 remember that, oh, a bar. 200 00:16:44,200 --> 00:16:44,830 Is. 201 00:16:46,440 --> 00:16:47,200 I. 202 00:16:48,260 --> 00:16:49,570 No JS. 203 00:16:50,940 --> 00:16:53,529 And two K's. 204 00:16:54,270 --> 00:16:56,418 That OB bar. 205 00:16:57,380 --> 00:17:00,070 Will be. Two I. 206 00:17:01,860 --> 00:17:07,770 Minus one J plus 4K. 207 00:17:09,490 --> 00:17:15,814 We want to know what's the magnitude of the vector AB bar. 208 00:17:17,050 --> 00:17:22,126 Just draw quick picture just to remind ourselves of how to get 209 00:17:22,126 --> 00:17:26,356 there. There's A and its position vector with respect to. 210 00:17:26,356 --> 00:17:31,432 Oh there's B with its position vector with respect to. If we're 211 00:17:31,432 --> 00:17:37,354 wanting a baby that's from there to there and so we can see that 212 00:17:37,354 --> 00:17:42,853 by going from A to B, we can go round AO plus OB. 213 00:17:44,340 --> 00:17:50,990 And so therefore, that is OB bar minus Oh, a bar. So that's what 214 00:17:50,990 --> 00:17:58,115 we need to do here. A bar must be OB bar minus oh, a bar. 215 00:17:59,130 --> 00:18:02,802 And all we do to do the subtraction is what you would 216 00:18:02,802 --> 00:18:05,556 do naturally, which is to subtract the respective bits 217 00:18:05,556 --> 00:18:09,228 so it's two I take away I. That's just an eye bar. 218 00:18:11,170 --> 00:18:17,425 Minus J takeaway no JS, so that's minus J 219 00:18:17,425 --> 00:18:22,985 Bar and 4K takeaway 2K. That's plus 2K. 220 00:18:24,190 --> 00:18:30,646 So now we have our vector AB bar. We can calculate its 221 00:18:30,646 --> 00:18:37,640 magnitude AB modulus of a bar that's just a be the length from 222 00:18:37,640 --> 00:18:45,172 A to B, and that's the square root of 1 squared plus minus one 223 00:18:45,172 --> 00:18:51,090 squared +2 squared altogether. That's 1 + 1 + 4 square 224 00:18:51,090 --> 00:18:54,318 root of 6, and the direction 225 00:18:54,318 --> 00:18:57,479 cosines. Our cause Alpha. 226 00:18:59,710 --> 00:19:04,372 That's The X coordinate over the modulus, so that's 227 00:19:04,372 --> 00:19:06,444 one over Route 6. 228 00:19:08,630 --> 00:19:14,810 Kohl's beta that's minus one over Route 6. the Y 229 00:19:14,810 --> 00:19:19,136 coordinate over the modulus and cause gamma. 230 00:19:20,680 --> 00:19:25,540 The zed coordinate over the modulus. 231 00:19:27,270 --> 00:19:31,470 Now this is a fairly standard calculation. The sort of 232 00:19:31,470 --> 00:19:33,570 calculation that it will be 233 00:19:33,570 --> 00:19:38,090 expected. You'll be able to do and simply be able to work your 234 00:19:38,090 --> 00:19:41,335 way through it very quickly. Very, very easily, so you have 235 00:19:41,335 --> 00:19:45,465 to be able to practice some of these. You have to be able to 236 00:19:45,465 --> 00:19:48,710 work with it very rapidly, very, very easily, but always keep 237 00:19:48,710 --> 00:19:49,890 this diagram in mind. 238 00:19:50,710 --> 00:19:57,750 That to get from A to B to form the vector AB bar, you go a 239 00:19:57,750 --> 00:19:59,950 obarr plus Obiba and so. 240 00:20:00,180 --> 00:20:07,590 It's the result, so to form a B it's Obi bar, take away OA bar.