1 00:00:01,260 --> 00:00:05,000 In this video for 3D geometry, we will be talking about two topics. 2 00:00:05,000 --> 00:00:08,960 One is the angle bisector of two planes and one is line of intersection of two planes. 3 00:00:08,960 --> 00:00:13,060 Both of the topics are important especially the line of intersection of two planes, and 4 00:00:13,060 --> 00:00:14,600 I will be spending some time on that. 5 00:00:14,600 --> 00:00:19,250 Angle bisector of two planes is a very easy topic, something you should remember while 6 00:00:19,250 --> 00:00:20,340 preparing for JEE. 7 00:00:20,340 --> 00:00:25,529 I will just give you the formula and I won’t be doing a problem but I will spending sometime 8 00:00:25,529 --> 00:00:28,824 on this (Line of Intersection of two planes), because this is a relatively important topic. 9 00:00:28,824 --> 00:00:33,530 So let us say we have a pair of planes. 10 00:00:33,530 --> 00:00:40,729 And you have to find the angle bisector. 11 00:00:40,729 --> 00:00:49,819 This is the angle bisector — plane which is the angle bisector of two planes. 12 00:00:49,819 --> 00:00:54,840 This angle is the same as this angle. 13 00:00:54,840 --> 00:00:59,760 You have been given equations for P1 and P2. 14 00:00:59,760 --> 00:01:04,459 You have find equation of angle bisector. 15 00:01:04,459 --> 00:01:12,860 If you think about this — easiest way to think about planes is using a notebook. 16 00:01:12,860 --> 00:01:16,970 I always recommend you do that if you are a little confused. 17 00:01:16,970 --> 00:01:21,110 So let us say I have two planes like this. 18 00:01:21,110 --> 00:01:22,430 Again, they are infinite. 19 00:01:22,430 --> 00:01:29,460 One plane will be the plane bisecting this angle. 20 00:01:29,460 --> 00:01:34,170 This plane can also be extended to this direction. 21 00:01:34,170 --> 00:01:38,960 Thus there can also be an angle bisector of this angle between the two planes. 22 00:01:38,960 --> 00:01:45,170 There are always two angles — one is acute and one is obtuse. 23 00:01:45,170 --> 00:01:49,090 You can find equations for both of the planes. 24 00:01:49,090 --> 00:01:51,190 The way you do it is very simple. 25 00:01:51,190 --> 00:01:56,720 I will describe the equations of both kind of planes. 26 00:01:56,720 --> 00:01:59,660 Again, there are two angles — acute and obtuse. 27 00:01:59,660 --> 00:02:00,810 How will you find the angles? 28 00:02:00,810 --> 00:02:02,410 Just take dot product between n1 and n2. 29 00:02:02,410 --> 00:02:03,890 You will get an angle — either obtuse or acute. 30 00:02:03,890 --> 00:02:10,478 If there is one angle, you can always subtract 180 - that to get the other angle. 31 00:02:10,478 --> 00:02:23,829 Let us say we have equations — this is just how you should remember to do 32 00:02:23,829 --> 00:02:24,829 this. 33 00:02:24,829 --> 00:02:29,670 I am currently writing d1 here so please take note of this. 34 00:02:29,670 --> 00:02:37,459 Generally I write d on the right hand side but this time it is here. 35 00:02:37,459 --> 00:02:50,829 Whenever you get something like this, ensure d1 and d2 are greater than 0. 36 00:02:50,829 --> 00:02:53,340 d1>0 , d2>0. 37 00:02:53,340 --> 00:02:57,840 Whatever equation you get, take constant to the left hand side, multiply by negative if 38 00:02:57,840 --> 00:02:59,810 there is a need to make this greater than 0. 39 00:02:59,810 --> 00:03:20,029 Once you have ensured that, check if a1a2 + b1b2 + c1c2 > 0. 40 00:03:20,029 --> 00:03:45,499 If it is greater than 0, then (a1x + b1y + c1z -d1)/(a1^2+b1^2+c1^2)^(1/2) = (a2x + b2y 41 00:03:45,499 --> 00:03:51,299 + c2z -d2)/(a2^2+b2^2+c2^2)^(1/2). 42 00:03:51,299 --> 00:03:57,959 This would be the angle bisector of the obtuse side. 43 00:03:57,959 --> 00:04:35,870 Similarly, let me just write it down so there is no confusion. 44 00:04:35,870 --> 00:04:45,170 This would be the angle for the acute side with negative sign in front. 45 00:04:45,170 --> 00:04:51,100 This is only when a1a2 + b1b2 + c1c2 > 0. 46 00:04:51,100 --> 00:04:57,170 If it is not greater than 0, vice-verse otherwise. 47 00:04:57,170 --> 00:05:06,630 The signs would reverse if this is less than 0. 48 00:05:06,630 --> 00:05:19,320 Just say opposite if a1a2 + b1b2 + c1c2 < 0. 49 00:05:19,320 --> 00:05:28,750 Think like this that whatever will be the sign of a1a2 + b1b2 + c1c2, that will be the 50 00:05:28,750 --> 00:05:32,379 sign of obtuse side. 51 00:05:32,379 --> 00:05:33,520 Thats how I remember. 52 00:05:33,520 --> 00:05:36,490 Again, you have to ensure that d1>0, d2>0. 53 00:05:36,490 --> 00:05:43,060 If is not a very conceptual thing and I can explain to you how this equation was derived, 54 00:05:43,060 --> 00:05:47,650 it is not hard to see. 55 00:05:47,650 --> 00:06:00,360 Thats not the important point here. 56 00:06:00,360 --> 00:06:05,160 If the question comes, just remember this thoroughly. 57 00:06:05,160 --> 00:06:08,920 It is literally like being careful with signs and calculation. 58 00:06:08,920 --> 00:06:11,910 So I am not doing a problem for this, you can do very easily yourself. 59 00:06:11,910 --> 00:06:17,900 You can construct two planes, and then find the angle bisector. 60 00:06:17,900 --> 00:06:19,560 I think its pretty straightforward. 61 00:06:19,560 --> 00:06:23,360 You just have to remember the formulas. 62 00:06:23,360 --> 00:06:27,790 Now let us go to the next part which is a very interesting topic — line of intersection 63 00:06:27,790 --> 00:06:28,900 of two planes. 64 00:06:28,900 --> 00:06:39,500 Let us say you have two planes P1 and P2. 65 00:06:39,500 --> 00:06:42,650 What is the line of intersection of two planes? 66 00:06:42,650 --> 00:06:52,310 That is the line which is — this line i.e. the line of intersection of two planes. 67 00:06:52,310 --> 00:06:53,830 If you have a notebook. 68 00:06:53,830 --> 00:06:59,970 The way you think about it — this line is the line of intersection of two planes. 69 00:06:59,970 --> 00:07:05,100 So just take a notebook and look at this line and that is the line of intersection. 70 00:07:05,100 --> 00:07:08,900 How would you find the equation of line? 71 00:07:08,900 --> 00:07:13,690 If you know P1 and P2, the question would be how to find this line L here. 72 00:07:13,690 --> 00:07:15,910 How would you define this. 73 00:07:15,910 --> 00:07:16,970 What all do you need to define a line? 74 00:07:16,970 --> 00:07:19,530 You need a parallel vector. 75 00:07:19,530 --> 00:07:24,730 You will have n1_vector and n2_vector. 76 00:07:24,730 --> 00:07:30,250 You need a point here, a_vector. 77 00:07:30,250 --> 00:07:33,980 And you need a parallel vector. 78 00:07:33,980 --> 00:07:35,610 What is the property of this parallel vector? 79 00:07:35,610 --> 00:07:41,430 Property of this parallel veto is that it is parallel to both P1 and P2 planes. 80 00:07:41,430 --> 00:07:45,850 Because it parallel to the line and the line is in both the planes, so it parallel to both 81 00:07:45,850 --> 00:07:48,180 planes P1 and P2. 82 00:07:48,180 --> 00:07:51,400 In other words, it is perpendicular to both n1 and n2. 83 00:07:51,400 --> 00:07:57,909 And as soon as I say this, it should flash into your mind that b_vector = n1 \cross n2. 84 00:07:57,909 --> 00:08:06,960 This is something straight out of the cross-product 85 00:08:06,960 --> 00:08:10,760 definition that b_vector = n1 \cross n2. 86 00:08:10,760 --> 00:08:17,200 Please recall the chapter of vectors if you are forgetting cross-product. 87 00:08:17,200 --> 00:08:22,760 So the first step is that you will fine n1 \cross n2 for getting b_vector. 88 00:08:22,760 --> 00:08:42,710 And second thing you would do is find a common point by putting one coordinate to be zero. 89 00:08:42,710 --> 00:08:48,300 I will solve a problem so that it will hopefully make it easier for you to understand. 90 00:08:48,300 --> 00:09:04,380 The equation for P1 that has been give to you is x+y+z=1. 91 00:09:04,400 --> 00:09:16,480 And the other plane has been give to you is x-y+z=1. 92 00:09:16,480 --> 00:09:22,320 First thing we have to do is that we have to find b_vector. What is n1_vector? n1 = i_cap 93 00:09:22,320 --> 00:09:33,120 + j_cap + k_cap. And n2_vector = i_cap - j_cap + k_cap. 94 00:09:33,120 --> 00:09:52,560 Now b_vector would be = |i j k; 1 1 1; 1 -1 1|. If you open this up, it will come 2i_cap 95 00:09:52,560 --> 00:10:03,180 - 2k_cap. You can check by taking a dot with both of them. This 2-2=0 and 2-2=0. This is 96 00:10:03,180 --> 00:10:05,700 a vector which is perpendicular to both n1 and n2. 97 00:10:05,700 --> 00:10:11,440 So this is the vector parallel to the line. 98 00:10:11,440 --> 00:10:21,520 Then you have to find the point. This was Step-1. Step-2 would be — we can put any 99 00:10:21,520 --> 00:10:26,560 coordinate to be zero. Let us put z=0 in both the planes. What we are doing is that we have 100 00:10:26,560 --> 00:10:29,760 to find a point. So we have to get the x coordinate, y coordinate and z coordinate. But you have 101 00:10:29,760 --> 00:10:34,240 two-equations and three variables. So we are putting one coordinate to be zero. If we put 102 00:10:34,240 --> 00:10:40,320 z=0, then you have two equations and two variables, and then you know the point. 103 00:10:40,320 --> 00:10:59,440 So we have x+y=1 and x-y=1. So that will give you x=1 and y=0. This will become x+y=1 and 104 00:10:59,480 --> 00:11:10,410 x-y=1. This becomes x=1, y=0. So the point A is (1,0,0). 105 00:11:10,410 --> 00:11:26,350 So equation of line L is i_cap + \lambda (2i_cap - 2k_cap). 106 00:11:26,350 --> 00:11:31,400 This line would lie in both plane 1 and plane 2. 107 00:11:31,400 --> 00:11:36,630 And you can also write it in the coordinate form. (x-1)/2 = (y-0)/0 = z/(-2). 108 00:11:36,630 --> 00:12:02,240 I hope that this made sense and gave you the idea of what it means. Line of intersection 109 00:12:02,240 --> 00:12:07,500 like a notebook is a line common to both the planes. It is perpendicular to both n1 and 110 00:12:07,500 --> 00:12:12,370 n2. So the parallel vector has to be n1 \cross n2. And then you can find the point by putting 111 00:12:12,370 --> 00:12:15,610 one coordinate to be zero. 112 00:12:15,610 --> 00:12:19,730 So this was the topic of angle bisector of two planes and line of intersection of two 113 00:12:19,730 --> 00:12:23,630 planes. The next video we will start mixing up line and planes, and points and planes, 114 00:12:23,630 --> 00:12:27,589 and things like that. I hope to see you in the next video. I hope you enjoyed this video. 115 00:12:27,589 --> 00:12:28,089 Thank you.