1 00:00:00,470 --> 00:00:01,310 ‫Welcome back. 2 00:00:01,760 --> 00:00:10,250 ‫So let's talk about the cross product applications now, one of the applications is actually a general 3 00:00:10,730 --> 00:00:12,970 ‫moment calculation formula. 4 00:00:13,550 --> 00:00:17,480 ‫So far we have computed moments in 2D, right. 5 00:00:18,200 --> 00:00:24,350 ‫Let's imagine a beam like in the assignment in the first part of the core series. 6 00:00:24,830 --> 00:00:31,470 ‫In that assignment, the beam initially hung on to courts like this. 7 00:00:31,850 --> 00:00:40,070 ‫This is the beam here and these are the two courts and this is the ceiling f g is the force of gravity 8 00:00:40,070 --> 00:00:46,010 ‫here that acts through the center of mass of the beam, which is here. 9 00:00:46,700 --> 00:00:51,290 ‫Then the left cord breaks and the beam starts falling down. 10 00:00:51,740 --> 00:00:56,720 ‫And you have to compute the moment about the point B right here. 11 00:00:57,380 --> 00:01:02,300 ‫When you did that, then everything happened on an X Y plane. 12 00:01:02,720 --> 00:01:07,010 ‫You had Y here and you had X here. 13 00:01:07,580 --> 00:01:11,180 ‫That is because we viewed our world in 2D. 14 00:01:11,570 --> 00:01:15,980 ‫However, in 3D, everything is much more general. 15 00:01:16,250 --> 00:01:26,030 ‫In 2D, the direction of the moment was either popping out of the plane towards you like this or away 16 00:01:26,030 --> 00:01:30,800 ‫from you like this, according to the right hand rule. 17 00:01:31,340 --> 00:01:37,180 ‫That is because that is how we measure the direction of rotations in the first place. 18 00:01:37,820 --> 00:01:45,890 ‫The rotation vector, whether it's angular velocity, angular acceleration or movement, it's vector 19 00:01:45,890 --> 00:01:50,450 ‫is perpendicular to the plane on which the rotation is happening. 20 00:01:50,990 --> 00:01:58,820 ‫So if something rotates like this, like you see here, then the vector of rotation is the red one here. 21 00:01:59,420 --> 00:02:03,260 ‫That's the direction of the rotation, the red vector. 22 00:02:03,740 --> 00:02:08,990 ‫Since moments cause rotation there, vectors are represented in the same way. 23 00:02:10,050 --> 00:02:19,710 ‫Now, in 3D, your moment vector can have infinite directions in a 3D space, you can imagine a sphere 24 00:02:19,710 --> 00:02:26,410 ‫here and the moment the vector can point in any direction on the sphere. 25 00:02:27,120 --> 00:02:38,040 ‫So instead of just saying moment equals distance times force like we did in the previous course in this 26 00:02:38,040 --> 00:02:43,760 ‫core series, we now have to go more generic in a 3D space. 27 00:02:44,100 --> 00:02:47,880 ‫We define a 3D position vector like this. 28 00:02:48,390 --> 00:03:03,360 ‫Our vector equals our X or Y and our Z transposed in a 3D force vector is like this affects F, Y and 29 00:03:03,570 --> 00:03:05,660 ‫Z transposed. 30 00:03:06,060 --> 00:03:13,380 ‫And of course this would be in meters and this would be in Newton's and the moment in 3D about a certain 31 00:03:13,380 --> 00:03:13,980 ‫point. 32 00:03:14,280 --> 00:03:25,320 ‫For example, Point O is calculated like this, the moment vector about the point O equals the position 33 00:03:25,320 --> 00:03:28,950 ‫vector cross the force vector. 34 00:03:29,640 --> 00:03:37,740 ‫That's the generic formula of moment calculation where you need the cross product and the cross product 35 00:03:37,740 --> 00:03:47,940 ‫here makes sure that only the component of the force vector that is perpendicular to the position vector 36 00:03:47,940 --> 00:03:49,440 ‫is taken into account. 37 00:03:49,830 --> 00:03:53,640 ‫Remember, that is what the cross product did. 38 00:03:53,850 --> 00:04:02,070 ‫It only multiplied the components of two vectors that were perpendicular to each other in the direction 39 00:04:02,070 --> 00:04:02,920 ‫of the moment. 40 00:04:02,940 --> 00:04:05,700 ‫Vector is perpendicular to both of them. 41 00:04:06,270 --> 00:04:13,230 ‫It's perpendicular to the position vector and it's also perpendicular to the force vector in 3D. 42 00:04:13,230 --> 00:04:15,060 ‫It looks something like this. 43 00:04:15,780 --> 00:04:17,910 ‫This is the point o here. 44 00:04:18,570 --> 00:04:20,910 ‫This is the position vector here. 45 00:04:21,690 --> 00:04:28,550 ‫The position vector could be the distance from the center of the drone to the propeller, for example. 46 00:04:29,100 --> 00:04:33,360 ‫So this is the center of the drone here and this is where the propeller is. 47 00:04:33,960 --> 00:04:42,000 ‫And then you also have the force vector here and this force vector could be the thrust of the propeller. 48 00:04:42,690 --> 00:04:53,340 ‫And so the drone experience is a moment m about its center that makes it angular accelerate about point. 49 00:04:53,340 --> 00:04:55,400 ‫Oh, like this. 50 00:04:56,130 --> 00:05:02,700 ‫So then of course means that this reference frame in white is the inertial reference frame and then 51 00:05:03,000 --> 00:05:11,640 ‫the body frame that is attached to the drone is this one, for example, small X that goes along this 52 00:05:11,640 --> 00:05:12,750 ‫position vector. 53 00:05:13,350 --> 00:05:14,850 ‫Be careful here though. 54 00:05:15,180 --> 00:05:30,720 ‫Order matters a krosby is not be cross a in fact A cross B equals minus B cross A and you saw it before 55 00:05:30,720 --> 00:05:32,310 ‫with the right hand rule as well. 56 00:05:32,940 --> 00:05:41,940 ‫To compute the moment you have to compute moment vector equals the position vector, cross the force 57 00:05:41,940 --> 00:05:44,640 ‫vector and not the other way around. 58 00:05:44,880 --> 00:05:52,460 ‫Force vector cross the position vector would be wrong, it would give you a wrong direction. 59 00:05:52,950 --> 00:05:58,250 ‫In fact it would give you a direction that is opposite to the right direction. 60 00:05:58,740 --> 00:06:08,760 ‫And again, remember, this green movement vector is perpendicular to both the position vector and to 61 00:06:08,760 --> 00:06:10,320 ‫the force vector. 62 00:06:10,320 --> 00:06:14,210 ‫If we shift this force vector here just for comparison. 63 00:06:14,640 --> 00:06:21,390 ‫So this green movement vector, as you can see, it forms a ninety degree angle with the position vector 64 00:06:21,900 --> 00:06:23,760 ‫and with the force vector. 65 00:06:24,090 --> 00:06:27,330 ‫And so the moment vector can also be written like this. 66 00:06:27,690 --> 00:06:34,430 ‫You have X and Y and then M, Z and then you transpose it. 67 00:06:34,800 --> 00:06:38,340 ‫So in the end, these are all three dimensional vectors here. 68 00:06:38,910 --> 00:06:49,470 ‫And so in a 2D case, it didn't matter whether you wrote F times the or the Times F that is because 69 00:06:49,470 --> 00:06:52,170 ‫we multiply two kilos with each other. 70 00:06:52,560 --> 00:07:00,750 ‫But in 3D, if you mess up the order, then your moment will be in the opposite direction because F 71 00:07:00,960 --> 00:07:09,330 ‫Cross R would give you minus M about the point o this would be the right. 72 00:07:09,410 --> 00:07:10,280 ‫Answer here. 73 00:07:10,970 --> 00:07:17,300 ‫So that has been one application of across product in the next video, we will talk about another one. 74 00:07:17,810 --> 00:07:18,380 ‫See you there.