1 00:00:13,610 --> 00:00:19,599 Now to introduce coordinates let me start with a simplest case of one-dimensional coordinate 2 00:00:19,599 --> 00:00:27,160 systems okay Let's assume that i have an infinitely long parallel road okay which is running along 3 00:00:27,160 --> 00:00:30,860 this way and there are houses situated along the road okay 4 00:00:30,860 --> 00:00:31,860 . 5 00:00:31,860 --> 00:00:35,570 You can imagine a highway or something where the road is long and there are houses situated 6 00:00:35,570 --> 00:00:41,989 along one edge of the road okay Now any road could be taken as a reference because this 7 00:00:41,989 --> 00:00:46,460 is an infinitely long road there are an infinite number of houses and you can take any house 8 00:00:46,460 --> 00:00:51,190 as a reference but it might be very easy for me to take my house as a reference okay So 9 00:00:51,190 --> 00:00:57,350 i will take my house as a reference and i call my house as house number 0 okay 10 00:00:57,350 --> 00:01:02,560 Of course i could have called this house as 150 or 200 it does not matter but 0 seems 11 00:01:02,560 --> 00:01:08,899 to be giving you a nice intuitive feeling that this is the origin this is the reference 12 00:01:08,899 --> 00:01:15,780 So the reference could be set to 0 and i take my house number to 0 okay Now what i start 13 00:01:15,780 --> 00:01:21,720 doing is there are different houses along the road i start labeling all these houses 14 00:01:21,720 --> 00:01:28,500 right So i label the houses 1 2 3 4 5 for example my friend's house happens to be 150th 15 00:01:28,500 --> 00:01:30,740 house from my house okay 16 00:01:30,740 --> 00:01:35,500 So this friend's house is 150th house from the reference house which is my house So i 17 00:01:35,500 --> 00:01:41,320 number them as 150 Then i have 151 152 and so on and it will continue like that There 18 00:01:41,320 --> 00:01:47,740 are houses to the left of my house as well So i will label them also 1 2 3 4 5 Now if 19 00:01:47,740 --> 00:01:52,729 i say house number 5 you would be confused Why would you be confused because i am not 20 00:01:52,729 --> 00:01:58,110 specifying whether this house number 5 is to the right of my house or to the left of 21 00:01:58,110 --> 00:01:59,780 my house right 22 00:01:59,780 --> 00:02:04,759 So once i fix the reference i also need to specify the direction in which i have to find 23 00:02:04,759 --> 00:02:10,310 this house So for example we have a mutual friend and i want to direct that mutual friend 24 00:02:10,310 --> 00:02:15,670 to one of my other friend's house right to my friend's house which is at 150 to the right 25 00:02:15,670 --> 00:02:23,920 of my house I have to specify go along right of my house and stop at 150th house okay 26 00:02:23,920 --> 00:02:30,349 So i have to specify both direction as well as the distance or the house number which 27 00:02:30,349 --> 00:02:37,810 i have to go So this is a one-dimensional system Now we can mathematically represent 28 00:02:37,810 --> 00:02:45,310 them that is we can convert this housing analogy to a coordinate system by drawing a line and 29 00:02:45,310 --> 00:02:52,090 we label this line as x axis and we call this point say my house as a reference house with 30 00:02:52,090 --> 00:02:53,400 a number 0 31 00:02:53,400 --> 00:02:59,709 I associate my house with 0 and then i start numbering all the other houses the house to 32 00:02:59,709 --> 00:03:04,510 distinguish between the house to the right and houses to the left i can say the houses 33 00:03:04,510 --> 00:03:10,900 to the right are positively numbered and houses to the left are negatively numbered So the 34 00:03:10,900 --> 00:03:15,989 house number one which was to the left of my house has now become minus one now i do 35 00:03:15,989 --> 00:03:19,170 not have to specify you know go to the right or go to the left 36 00:03:19,170 --> 00:03:24,870 I just have to specify minus 150th house or plus 150th house may be there is a house in 37 00:03:24,870 --> 00:03:31,020 between 150 and 151 this house could be 1505 So i could say “hey go to the location of 38 00:03:31,020 --> 00:03:41,700 1505” May be there is a landmark at 2035 or 2038 right So i could then say go to the 39 00:03:41,700 --> 00:03:50,120 landmark 2038 or go to the landmark at minus 1536 right So i can specify that and x is 40 00:03:50,120 --> 00:03:53,220 the axis that i would be calling okay 41 00:03:53,220 --> 00:04:00,530 So what this house analogy has told you is that i can associate on this road any point 42 00:04:00,530 --> 00:04:07,710 right i can specify that point by giving it a number So i have converted or i have found 43 00:04:07,710 --> 00:04:15,810 the method in which i can locate a point in space by a number I can specify that point 44 00:04:15,810 --> 00:04:21,349 in space by a number For example the position of the nth house can be specified by giving 45 00:04:21,349 --> 00:04:28,190 a number n Now let us say i have a friend okay who does not agree to consider my house 46 00:04:28,190 --> 00:04:31,830 as a reference i mean everyone would like to consider their house as the reference 47 00:04:31,830 --> 00:04:37,259 So i have a friend who wants to consider his house which happens to be house number 1 as 48 00:04:37,259 --> 00:04:44,189 the reference So what he or she does is that they keep this house 1 as the reference house 49 00:04:44,189 --> 00:04:50,150 okay So in their coordinate system or in their method of measuring the locations or specifying 50 00:04:50,150 --> 00:04:56,129 the locations of houses one becomes 0 n becomes n minus one 51 00:04:56,129 --> 00:05:01,490 Does that mean that the nth house has actually physically changed No the house is situated 52 00:05:01,490 --> 00:05:08,159 where it was it's number that we are specifying has become different because in my mind 0 53 00:05:08,159 --> 00:05:15,249 is the reference house and for his coordinate system 1 is the reference or 1 is the 0 for 54 00:05:15,249 --> 00:05:20,020 that one So because the origins are different the number that we come up with are different 55 00:05:20,020 --> 00:05:22,360 but physically this is the same house 56 00:05:22,360 --> 00:05:27,409 The house has not changed just because i have decided or my friend decided that the origin 57 00:05:27,409 --> 00:05:32,059 needs to be different okay So i hope you get the difference between specifying a coordinate 58 00:05:32,059 --> 00:05:38,249 system and specifying a number okay This is closely related to the fact that vectors we 59 00:05:38,249 --> 00:05:42,119 will soon see how to connect vectors here So this is closely related to the fact that 60 00:05:42,119 --> 00:05:47,839 vectors are independent whereas the number that we associate with a vector is dependent 61 00:05:47,839 --> 00:05:53,479 on the coordinate system that i choose 62 00:05:53,479 --> 00:05:57,749 So now how do we associate vectors to this coordinate system we have already established 63 00:05:57,749 --> 00:06:03,960 a coordinate system okay and let's say the coordinate system that i have established 64 00:06:03,960 --> 00:06:05,899 should allow me to measure distances . 65 00:06:05,899 --> 00:06:11,270 So i do not want to say the position of 150th house i want to say “hey my friend's house 66 00:06:11,270 --> 00:06:18,159 which is at 150 is actually at 150 kilometers” or it is at 150 meters or it is at 150 centimeters 67 00:06:18,159 --> 00:06:25,669 okay So there is a landmark at 2038 kilometers so i want to specify distances out there but 68 00:06:25,669 --> 00:06:32,500 you will see what i did over here i had a distance of kilometer meter centimeter How 69 00:06:32,500 --> 00:06:39,319 was this distance or how was this unit of measure formulated Let's lookay at this I 70 00:06:39,319 --> 00:06:44,460 have all these houses it is again see that i am using my coordinate systems 71 00:06:44,460 --> 00:06:50,710 So i can use my coordinate system and say i measure the distance from my house to my 72 00:06:50,710 --> 00:06:57,080 next house along the right which is say 1 and i call this as the unit distance So all 73 00:06:57,080 --> 00:07:03,759 other distances along x axis are now measured in units of this distance okay So this is 74 00:07:03,759 --> 00:07:12,249 1 unit so 150th house will be 150 units away from 0 again there is no specific reason why 75 00:07:12,249 --> 00:07:15,749 we have to specify this as 1 unit distance 76 00:07:15,749 --> 00:07:22,399 I could specify the distance between midway from my house to my friend's house or to my 77 00:07:22,399 --> 00:07:28,729 neighbor's house as 1 unit then the 150th house which is my friend's house will be at 78 00:07:28,729 --> 00:07:37,539 a distance of 150 into two which is 300 units if i half the unit So it is clearly my control 79 00:07:37,539 --> 00:07:42,409 as to what i should call as a unit and we pick whatever the unit that would correspond 80 00:07:42,409 --> 00:07:48,339 to us and we just stick to that unit and this kilometer meter centimeter are all the choices 81 00:07:48,339 --> 00:07:49,339 that we make 82 00:07:49,339 --> 00:07:53,919 So if i take 1 kilometer as the unit then all the other distances can be measured and 83 00:07:53,919 --> 00:08:00,169 will be measured as some multiple of kilometer That multiple need not be integral it need 84 00:08:00,169 --> 00:08:08,129 not always be 1 kilometer 2 kilometer 3 kilometer Our landmark could be at 23 kilometers okay 85 00:08:08,129 --> 00:08:12,619 So i hope that distinction is very well understood by you 86 00:08:12,619 --> 00:08:19,199 Now this is a coordinate system along to the right there is x axis i have also obtained 87 00:08:19,199 --> 00:08:25,789 the units on which i am going to measure now how do i define a vector Consider the nth 88 00:08:25,789 --> 00:08:33,969 house okay I can specify the nth house by giving its distance from the origin origin 89 00:08:33,969 --> 00:08:39,430 being my house which i am taking as the origin or the reference i can give the position of 90 00:08:39,430 --> 00:08:50,670 this nth house as n units away from 0 I can equivalently draw a vector whose length is 91 00:08:50,670 --> 00:08:57,800 n and whose origin is at 0 and which terminates at n 92 00:08:57,800 --> 00:09:02,379 Now you can imagine that i am first removing the coordinate system i just have a vector 93 00:09:02,379 --> 00:09:09,110 which is lying horizontally of this length 0 to n okay I mean of length this line and 94 00:09:09,110 --> 00:09:14,470 now if i want to attach a coordinate system i can simply imagine a coordinate system over 95 00:09:14,470 --> 00:09:19,850 here in which the origin coincides with the tail of the blue vector okay Which is what 96 00:09:19,850 --> 00:09:28,470 the vector n is okay So this vector n can be represented as n times x hat 97 00:09:28,470 --> 00:09:35,790 What is this funny lookaying x hat over here X hat tells me a unit vector along the direction 98 00:09:35,790 --> 00:09:43,060 of x What is a unit vector It is a vector which has unit magnitude and points in the 99 00:09:43,060 --> 00:09:46,899 direction of the coordinate system Here i have a one-dimensional coordinate system along 100 00:09:46,899 --> 00:09:53,430 x so this is pointing along x direction has a magnitude of one so this unit vector have 101 00:09:53,430 --> 00:09:56,940 indicated by this orange colored vector okay 102 00:09:56,940 --> 00:10:02,550 So orange colored line arrow is a unit vector and if i take a unit vector and multiply it 103 00:10:02,550 --> 00:10:08,310 by a number n then i get the vector n which has the magnitude of n and it will not be 104 00:10:08,310 --> 00:10:14,410 pointing along the x direction okay So now i have shown you two things one there is a 105 00:10:14,410 --> 00:10:19,060 position which is the location of the space along the x axis that can be specified by 106 00:10:19,060 --> 00:10:20,750 giving it a number 107 00:10:20,750 --> 00:10:24,829 So if you imagine that there is no vector over here if you imagine that there is no 108 00:10:24,829 --> 00:10:30,310 vector here but just a number n that is sitting here on the x axis then the location of this 109 00:10:30,310 --> 00:10:37,000 point where n is sitting is nm okay You can have another position which would be m which 110 00:10:37,000 --> 00:10:43,440 would be along the x axis Now i have a vector with no specific reference to the coordinate 111 00:10:43,440 --> 00:10:48,870 system However i can introduce a coordinate system and say that this vector can be represented 112 00:10:48,870 --> 00:10:55,370 as n x hat where n represents the magnitude of the vector and x hat represents the direction 113 00:10:55,370 --> 00:10:56,389 of the unit vector 114 00:10:56,389 --> 00:11:01,480 So in this case the unit vectors can only be directed along x Now suppose i want to 115 00:11:01,480 --> 00:11:06,639 represent a vector which has same magnitude n but it will be pointing in the negative 116 00:11:06,639 --> 00:11:11,699 x direction how would i represent that All you have to do is you have to represent this 117 00:11:11,699 --> 00:11:18,189 as minus n x hat You could of course take a unit vector which will be pointing in the 118 00:11:18,189 --> 00:11:22,509 negative x direction you can call that as some minus x hat 119 00:11:22,509 --> 00:11:29,560 Then i can multiply that by the required magnitude say the value of n and then obtain minus n 120 00:11:29,560 --> 00:11:34,769 x hat What we preferred to do is that we keep the unit vectors fixed If i want to refer 121 00:11:34,769 --> 00:11:41,220 to a negative direction i simply multiply that one by a sign number if the sign number 122 00:11:41,220 --> 00:11:45,970 is positive it will be pointing in the same direction as the unit vector if the sign number 123 00:11:45,970 --> 00:11:51,430 is negative right the multiplier is negative then it will be pointing in the opposite direction 124 00:11:51,430 --> 00:11:53,279 okay . 125 00:11:53,279 --> 00:11:58,220 So this was the one-dimensional coordinate system a vector n So let me recap this a vector 126 00:11:58,220 --> 00:12:04,040 n which is starting at origin o and ending at a point n is given by n x hat Please note 127 00:12:04,040 --> 00:12:07,550 the notation that we are going to use This will be the notation that we will be using 128 00:12:07,550 --> 00:12:14,290 throughout our study okay Here the number n which can be positive or negative okay If 129 00:12:14,290 --> 00:12:18,879 we associate and sign along with this one if i do not have then n is a positive number 130 00:12:18,879 --> 00:12:24,700 So n is a magnitude of the vector n and x is the unit vector along x axis If i want 131 00:12:24,700 --> 00:12:29,560 to represent a vector which is acting along minus x direction or which is along the negative 132 00:12:29,560 --> 00:12:35,759 x direction then i have to assign sign here okay I have to say minus n x hat which will 133 00:12:35,759 --> 00:12:40,620 then point to a vector whose magnitude is n that is whose length is n but it will be 134 00:12:40,620 --> 00:12:45,110 acting in the minus x direction negative x direction okay 135 00:12:45,110 --> 00:12:50,610 If i take a unit vector and then multiply it by any number i am essentially stretching 136 00:12:50,610 --> 00:12:56,069 the unit vector okay If the multiplier is less than 1 then i am compressing the vector 137 00:12:56,069 --> 00:13:00,319 If the multiplier is equal to minus 1 then i am taking the vector and changing the direction 138 00:13:00,319 --> 00:13:07,550 from plus to minus okay Now let us go to two-dimensional coordinate system Whatever fun we had to have 139 00:13:07,550 --> 00:13:13,920 with one-dimensions is all over now let us go to two-dimensional scenarios okay Now here 140 00:13:13,920 --> 00:13:19,799 again the notion of coordinate system as well as the vector will be different okay 141 00:13:19,799 --> 00:13:20,799 . 142 00:13:20,799 --> 00:13:24,639 So you can have a vector which is independent of the coordinate system and then you choose 143 00:13:24,639 --> 00:13:29,550 a coordinate system such that the numbers that you get will be unambiguously defining 144 00:13:29,550 --> 00:13:34,670 a vector and it will be simpler to work with So you have an unambiguous definition of a 145 00:13:34,670 --> 00:13:41,079 vector in the chosen coordinate system as well as an easier method of handling the numbers 146 00:13:41,079 --> 00:13:47,060 okay Just to bring out the distinction lookay at this figure you have a vector which i am 147 00:13:47,060 --> 00:13:52,089 calling as vector a which is given by red colored arrow over here 148 00:13:52,089 --> 00:13:57,629 This red colored arrow as in the previous vectors will have 2 points there is a starting 149 00:13:57,629 --> 00:14:02,879 or the initial points Because this is a two-dimensional coordinate system you might expect that every 150 00:14:02,879 --> 00:14:09,269 point in the position will be specified positioning space will be specified by 2 numbers So this 151 00:14:09,269 --> 00:14:17,050 origin point of the vector a will be specified by 2 numbers x i and y i Similarly the end 152 00:14:17,050 --> 00:14:23,839 point of the vector will be specified by 2 numbers which are x f and y f okay 153 00:14:23,839 --> 00:14:29,980 What are these x and y These are the 2 lines or the directions which are needed to completely 154 00:14:29,980 --> 00:14:35,480 specify any point in the coordinate system In the two-dimensional system i have to specify 155 00:14:35,480 --> 00:14:42,589 the location of any point by 2 numbers One along x and one along y Imagine that i have 156 00:14:42,589 --> 00:14:45,870 this plane and then there are houses scattered all over 157 00:14:45,870 --> 00:14:49,959 Now if i want to reach a house which is at some particular point in this space over here 158 00:14:49,959 --> 00:14:56,580 i can say “hey go along horizontally 10 units of houses” or "10th house and from 159 00:14:56,580 --> 00:15:05,170 that house go vertically to 20th house to reach the house which is numbered 10 20" okay 160 00:15:05,170 --> 00:15:11,160 This is the meaning of giving 2 coordinates and you can notice that the directions horizontal 161 00:15:11,160 --> 00:15:13,220 and vertical are perpendicular to each other 162 00:15:13,220 --> 00:15:17,440 We will talk a little bit about perpendicular the requirement of perpendicularity later 163 00:15:17,440 --> 00:15:22,899 okay At any point we can be considered as origin so we consider this point o as the 164 00:15:22,899 --> 00:15:28,709 origin here So anything to the increasing a side of x will be positive this would be 165 00:15:28,709 --> 00:15:35,500 positive if you go below this line you will be going minus y if you go to the left you 166 00:15:35,500 --> 00:15:41,079 will be moving along minus x direction okay So any point in this two-dimensional space 167 00:15:41,079 --> 00:15:46,279 or a plane can be specified by 2 numbers okay 168 00:15:46,279 --> 00:15:51,829 So here is a vector a with its initial and final points Here is a coordinate system that 169 00:15:51,829 --> 00:15:57,230 my friend has come up with Now every time i specify the vector a i have to specify both 170 00:15:57,230 --> 00:16:02,519 the initial as well as the final points Now this might become little tricky okay because 171 00:16:02,519 --> 00:16:07,139 every vector needs to be specified by 4 numbers now one for the starting point and one for 172 00:16:07,139 --> 00:16:14,579 the end point On the other hand it would be wonderful if i take the same origin okay for 173 00:16:14,579 --> 00:16:16,389 all the vectors 174 00:16:16,389 --> 00:16:23,040 What am i getting by doing so I am eliminating this need of specifying this x i and y i all 175 00:16:23,040 --> 00:16:29,089 the time I do not have to specify x i and y i all the time If i choose my coordinate 176 00:16:29,089 --> 00:16:35,831 system as coinciding with the tail or the origin of this vector but the coordinate system 177 00:16:35,831 --> 00:16:40,680 does not seem to be coinciding What is the solution I cannot move the vector of course 178 00:16:40,680 --> 00:16:45,560 i can move the vector but normally what we think of is we do not move the vector we move 179 00:16:45,560 --> 00:16:46,660 the coordinate system 180 00:16:46,660 --> 00:16:50,779 There are 2 equivalent ways of doing the same thing Either you move the vector parallelly 181 00:16:50,779 --> 00:16:56,410 until you reach the origin of the coordinate system okay We can do that one or we simply 182 00:16:56,410 --> 00:17:01,449 move the coordinate system until you reach the origin and moving the coordinate system 183 00:17:01,449 --> 00:17:06,160 not only means parallelly moving them you can also twist and rotate the coordinate systems 184 00:17:06,160 --> 00:17:09,449 okay Some of these things will become important later 185 00:17:09,449 --> 00:17:13,930 At this point let us not clutter too much about that so what i am going to do is that 186 00:17:13,930 --> 00:17:21,130 i am going to redraw my lines y and x such that the origin of the coordinate system o 187 00:17:21,130 --> 00:17:27,370 right coincides with the origin of the vector a Thus a vector a will now be specified only 188 00:17:27,370 --> 00:17:35,659 by 2 points x and y Why because this 0 and 0 the origin is implicitly understood If i 189 00:17:35,659 --> 00:17:42,149 had another vector i could draw a line from the origin to the other point and that other 190 00:17:42,149 --> 00:17:45,730 vector could also be specified by only 2 numbers 191 00:17:45,730 --> 00:17:51,429 This is the advantage of coinciding or making your origin of the coordinate system coincide 192 00:17:51,429 --> 00:17:57,130 with the tail of all the vectors How about the vector difference Well you have to wait 193 00:17:57,130 --> 00:18:03,220 for a few slides to see how to define the sum and addition of vectors when you have 194 00:18:03,220 --> 00:18:10,049 2 different vectors here okay But for now please remember that you can draw a vector 195 00:18:10,049 --> 00:18:15,650 from the origin to any other point in the two-dimensional space All vectors are referred 196 00:18:15,650 --> 00:18:19,100 with respect to origin okay . 197 00:18:19,100 --> 00:18:24,399 So you have a vector a which is referenced with the origin o here and let's say this 198 00:18:24,399 --> 00:18:32,840 vector a is terminating at a point 25 and 3 What is this 25 and 3 mean If you forget 199 00:18:32,840 --> 00:18:39,460 the vector a this simply gives you the location of the point 25 and 3 in the two-dimensional 200 00:18:39,460 --> 00:18:46,240 space that is described by this vector x and y correct So i have a vector x and y and all 201 00:18:46,240 --> 00:18:49,990 points are specified by 2 coordinates 202 00:18:49,990 --> 00:18:58,809 So in this particular case the vector a goes from the origin and terminates at 25 and 3 203 00:18:58,809 --> 00:19:07,929 Now to go from o to a i can of course go along the red line or i can go along the horizontal 204 00:19:07,929 --> 00:19:13,169 direction until i reach this point You can see that where i am reaching this one and 205 00:19:13,169 --> 00:19:15,940 then continue in the vertical direction 206 00:19:15,940 --> 00:19:22,789 Now if you remember how we added two vectors you had one vector tail origin to that you 207 00:19:22,789 --> 00:19:27,169 added another vector that is you brought in another vector by parallelly translating it 208 00:19:27,169 --> 00:19:32,190 such that the tail of the second vector coincided with the head of the first vector Now i have 209 00:19:32,190 --> 00:19:38,050 a tail of this vector which is now getting added to the vector which is horizontal So 210 00:19:38,050 --> 00:19:41,130 i have a horizontal vector then a vertical vector 211 00:19:41,130 --> 00:19:47,000 The sum of these 2 vectors is obviously the vector o a which you can see very clearly 212 00:19:47,000 --> 00:19:51,630 over here What are the 2 vectors that i am adding The horizontal vector which is given 213 00:19:51,630 --> 00:19:57,970 by this blue line and then a vertical vector which is given by this blue line What is the 214 00:19:57,970 --> 00:20:04,169 magnitude of the horizontal vector It is exactly this 25 and in which direction does it point 215 00:20:04,169 --> 00:20:13,740 It points along the x direction So i can represent this vector itself as 25 x hat right This 216 00:20:13,740 --> 00:20:19,919 vector is 25 x hat and this is precisely the vector that is giving you this horizontal 217 00:20:19,919 --> 00:20:20,919 vector 218 00:20:20,919 --> 00:20:26,279 So this horizontal vector is characterized by x hat a x where x hat is the direction 219 00:20:26,279 --> 00:20:32,059 of the vector which is along the x axis and a x is the magnitude of that vector that is 220 00:20:32,059 --> 00:20:38,750 of the horizontal vector which is 25 so this is another way of saying that you move 25 221 00:20:38,750 --> 00:20:44,790 units along x and then move 3 units along y and if you move along y you are actually 222 00:20:44,790 --> 00:20:50,559 creating a vector here which again is nothing but a parallelly transmitted vector y hat 223 00:20:50,559 --> 00:20:55,179 a y and what is a y here A y is 3 224 00:20:55,179 --> 00:21:02,500 So this original vector o a has been resolved or decomposed into 2 vectors the sum of 2 225 00:21:02,500 --> 00:21:08,000 vectors both this vectors are so one of the vectors is along x the other vector is along 226 00:21:08,000 --> 00:21:13,169 y These two themselves are perpendicular to each other You can see that from the graph 227 00:21:13,169 --> 00:21:21,490 the vector a x x hat is perpendicular to the vector a y y hat okay Similarly as we defined 228 00:21:21,490 --> 00:21:26,600 a unit vector for the x axis i can define a unit vector for the y axis and i have actually 229 00:21:26,600 --> 00:21:31,850 used that definition of unit vector of the y axis when i am writing this vector y hat 230 00:21:31,850 --> 00:21:33,520 a y okay 231 00:21:33,520 --> 00:21:39,800 So the vector o a has been resolved into 2 components that is sum of two vectors one 232 00:21:39,800 --> 00:21:45,529 vector along x and the other vector along y So i have two vectors one along x and one 233 00:21:45,529 --> 00:21:52,580 along y The sum of these two is giving me the vector o a okay Keep this in mind okay 234 00:21:52,580 --> 00:21:53,580 . 235 00:21:53,580 --> 00:21:57,390 Now we move from two-dimensional coordinate system to three-dimensional coordinate system 236 00:21:57,390 --> 00:22:02,340 Now this is why electromagnetic is sometimes thought to be very abstract mathematical and 237 00:22:02,340 --> 00:22:07,200 at of subject because you have to work with three-dimensions However your work will be 238 00:22:07,200 --> 00:22:12,269 simplified if you understand the coordinate systems and if you choose an appropriate coordinate 239 00:22:12,269 --> 00:22:14,070 system for your problem okay 240 00:22:14,070 --> 00:22:20,179 We will see the tragic consequences of choosing a bad coordinate system to solve a particular 241 00:22:20,179 --> 00:22:25,850 electromagnetic problem sometime later For now let us try to understand the three-dimensional 242 00:22:25,850 --> 00:22:30,510 coordinate system If you have followed the discussion of one-dimension and two-dimensional 243 00:22:30,510 --> 00:22:35,149 coordinate system extending this to three-dimensions should not be a problem except that it will 244 00:22:35,149 --> 00:22:40,460 be little mentally taxing in visualizing the three-dimensional vector 245 00:22:40,460 --> 00:22:46,070 If you go back to two-dimensional case you had two lines may be we can take this as the 246 00:22:46,070 --> 00:22:50,769 two-dimensional coordinate system example so you had 2 lines which were perpendicular 247 00:22:50,769 --> 00:22:55,309 to each other and these 2 lines are mutually perpendicular to each other and you define 248 00:22:55,309 --> 00:23:01,260 a unit vector along one and you define another unit vector along another axis So you have 249 00:23:01,260 --> 00:23:06,159 a unit vector x hat you have a unit vector y hat okay 250 00:23:06,159 --> 00:23:12,080 So any point was actually given by intersection of 2 lines For example you lookay at this 251 00:23:12,080 --> 00:23:18,440 point 25 3 this point at which the vector a is terminating is actually the intersection 252 00:23:18,440 --> 00:23:25,080 of 2 lines This line this horizontal line that is shown here is the horizontal line 253 00:23:25,080 --> 00:23:32,769 which corresponds to the constant value of x okay So this line is x is equal to 25 Whereas 254 00:23:32,769 --> 00:23:38,809 this horizontal line that i have shown here dash is the line y is equal to 3 okay 255 00:23:38,809 --> 00:23:45,010 This line is x is equal to 25 you can actually stretch this along this direction and no matter 256 00:23:45,010 --> 00:23:49,669 where you are on this point the value of x will always be the same The value of x along 257 00:23:49,669 --> 00:23:56,190 this dashed vertical line will always be same and it will be equal to 25 The value of y 258 00:23:56,190 --> 00:24:02,230 along this horizontal line will be the same and it will be equal to 3 Only thing is x 259 00:24:02,230 --> 00:24:07,850 will be changing in the horizontal line whereas x will be constant here and y will be changing 260 00:24:07,850 --> 00:24:10,090 as you move up and down okay 261 00:24:10,090 --> 00:24:15,789 So any point in two-dimensional coordinate system was represented as intersection of 262 00:24:15,789 --> 00:24:23,360 2 points Now on a three-dimensional coordinate system you are lookaying at intersection of 263 00:24:23,360 --> 00:24:30,840 mutually perpendicular axes which form 3 planes okay What is this plane See now lookay at 264 00:24:30,840 --> 00:24:36,559 this x is equal to 0 plane i have 3 mutually perpendicular axes that means that axis x 265 00:24:36,559 --> 00:24:43,090 will be perpendicular to axis y axis y will be perpendicular to axis z okay Of course 266 00:24:43,090 --> 00:24:49,909 axis z is perpendicular to axis x Lookay at x is equal to 0 plane okay 267 00:24:49,909 --> 00:24:56,090 All points on this plane have the x value which is equal to 0 and if you lookay at this 268 00:24:56,090 --> 00:25:03,840 horizontal plane this is z is equal to 0 plane because all points on this plane have a constant 269 00:25:03,840 --> 00:25:10,500 value of z which is 0 in this case Similarly you have another plane which is y is equal 270 00:25:10,500 --> 00:25:16,890 to 0 you can go up which means z is changing you can go along the direction of x which 271 00:25:16,890 --> 00:25:22,260 means x is changing but you are not moving away from the plane which means y is constant 272 00:25:22,260 --> 00:25:27,000 So you have y is equal to 0 or y is equal to constant In this case the constant is 0 273 00:25:27,000 --> 00:25:31,909 plane You have x is equal to 0 plane you have z is equal to 0 plane and the intersection 274 00:25:31,909 --> 00:25:39,360 point of all the 3 planes defines the origin Now if i want to specify a general x y z point 275 00:25:39,360 --> 00:25:43,630 how do i specify Well you have to put up 3 mutually perpendicular planes there you have 276 00:25:43,630 --> 00:25:49,210 to put up X is equal to constant plane y is equal to constant plane and z is equal to 277 00:25:49,210 --> 00:25:50,210 constant plane 278 00:25:50,210 --> 00:25:56,480 So you have to put up 3 mutually perpendicular axes composed of 3 different planes so as 279 00:25:56,480 --> 00:26:01,530 to specify any point So clearly any point in a three-dimensional space will have to 280 00:26:01,530 --> 00:26:09,130 be specified by 3 points okay Note the direction of the vector arrows over here i have a vector 281 00:26:09,130 --> 00:26:13,110 x which is increasing in this direction i have a vector y which is increasing in this 282 00:26:13,110 --> 00:26:16,880 direction and i have a vector z which is increasing in this direction okay 283 00:26:16,880 --> 00:26:24,220 The specific directions were chosen to satisfy what is called as right-handed coordinate 284 00:26:24,220 --> 00:26:29,019 system such that the resultant coordinate system is right-handed coordinate system What 285 00:26:29,019 --> 00:26:33,680 is right-handed coordinate system Right-handed coordinate system is shown here You imagine 286 00:26:33,680 --> 00:26:39,620 yourself as using your thumb finger to point along the z direction okay 287 00:26:39,620 --> 00:26:45,460 So if you imagine yourself as using the thumbed point along the z direction you will see that 288 00:26:45,460 --> 00:26:50,360 the index finger will be pointing along the x and the curved fingers will be pointing 289 00:26:50,360 --> 00:26:56,880 along y direction okay So you have thumb pointing along z direction and the index finger pointing 290 00:26:56,880 --> 00:27:02,460 along the x direction and the curling is pointing along the y direction The curled finger would 291 00:27:02,460 --> 00:27:04,260 be pointing along the y direction 292 00:27:04,260 --> 00:27:10,679 So in another words if you go from x to y you will be going up along the z direction 293 00:27:10,679 --> 00:27:15,299 You can imagine a different coordinate system in which if you go from x to y you will be 294 00:27:15,299 --> 00:27:21,519 moving along the minus z direction okay Minus z with respect to this particular z i am talking 295 00:27:21,519 --> 00:27:26,309 about that would be a coordinate system that would be lookaying like this okay It would 296 00:27:26,309 --> 00:27:30,830 be an inverted coordinate system and sometimes called as the left-handed coordinate system 297 00:27:30,830 --> 00:27:31,830 . 298 00:27:31,830 --> 00:27:35,470 Let us lookay at how to specify the points on a three-dimensional coordinate system To 299 00:27:35,470 --> 00:27:42,250 specify a point which is x y and z okay i need 3 planes as i have just told you i need 300 00:27:42,250 --> 00:27:50,529 3 planes So if you lookay at this plane this plane should be erected at x is equal to some 301 00:27:50,529 --> 00:27:57,289 constant okay whatever the value that is given to me and this plane you can see here is to 302 00:27:57,289 --> 00:28:02,840 be erected at y is equal to constant So whatever the value of y that i need to specify i have 303 00:28:02,840 --> 00:28:05,509 to put up a plane here 304 00:28:05,509 --> 00:28:09,690 The intersection of the planes x is equal to constant and y is equal to constant will 305 00:28:09,690 --> 00:28:15,990 give me 2 coordinates x and y To get the third coordinate i have to now put up a plane which 306 00:28:15,990 --> 00:28:22,110 is z is equal to constant okay So 2 planes will give me 2 coordinates the third plane 307 00:28:22,110 --> 00:28:27,910 will determine the final point or the final coordinate of the location in space Again 308 00:28:27,910 --> 00:28:33,090 you have three mutually perpendicular directions which is one vector pointing along x one pointing 309 00:28:33,090 --> 00:28:36,370 along y and one pointing along z direction 310 00:28:36,370 --> 00:28:43,730 For example consider point p and q point p is -2 2 and 3 Which means that i have to put 311 00:28:43,730 --> 00:28:50,220 up a plane at x is equal to -2 x is equal to -2 occurs beyond this For example this 312 00:28:50,220 --> 00:28:56,880 is the x is equal to -2 and then i have to move 2 point 2 on y axis so i can actually 313 00:28:56,880 --> 00:29:03,549 move along the x line to the y axis to get me this point okay This is equivalent of putting 314 00:29:03,549 --> 00:29:08,870 up a plane at y is equal to 2 and plane at x is equal to -2 the intersection of those 315 00:29:08,870 --> 00:29:11,529 two will be at -2 and 2 316 00:29:11,529 --> 00:29:18,490 Now if you move up because this is 3 if you move up to the point 3 or the plane where 317 00:29:18,490 --> 00:29:25,899 z is equal to 3 you will end up with point p So to get to point p you move -2 along x 318 00:29:25,899 --> 00:29:34,950 2 along y and 3 units along z Similarly to get to y which is 3 -2 and 2 you have to first 319 00:29:34,950 --> 00:29:39,470 move along x is equal to 3 that is to move to the plane x is equal to 3 So i moved to 320 00:29:39,470 --> 00:29:41,460 this plane x is equal to 3 321 00:29:41,460 --> 00:29:46,820 On this plane y and z values could be anything but y is equal to 2 is given So i need to 322 00:29:46,820 --> 00:29:51,600 now move to y is equal to minus 2 planes So i need to setup y is equal to minus 2 plane 323 00:29:51,600 --> 00:29:57,389 Now when i bring these 2 planes together i am at a point which is 3 and 2 Now i have 324 00:29:57,389 --> 00:30:03,639 to move to point 2 along z so which means i have to erect a plane at z is equal to 2 325 00:30:03,639 --> 00:30:10,809 So i have moved to point 2 So this q is defined by 3 numbers 3 - 2 and 2 okay 326 00:30:10,809 --> 00:30:16,740 A different notation is to just give you the vector i mean just give you the point p and 327 00:30:16,740 --> 00:30:22,169 give the coordinates of the point So p and in brackets you give the coordinates say in 328 00:30:22,169 --> 00:30:28,749 this case it is -2 2 and 3 and for q it would be 3 -2 and 2 okay 329 00:30:28,749 --> 00:30:29,749 . 330 00:30:29,749 --> 00:30:35,789 So this is how to specify locations on a three-dimensional coordinate system I hope that you have understood 331 00:30:35,789 --> 00:30:43,610 this particular point and i suggest that you try to graph more number of points to become 332 00:30:43,610 --> 00:30:45,659 familiar with this particular coordinate system