1 00:00:00,750 --> 00:00:08,560 ‫Welcome back, I hope you tried the exercise yourself before watching this video, so we rotated pie 2 00:00:08,640 --> 00:00:20,340 ‫over to Radiance about the inertial Z axis and this is what we got again, p, q, r, r, one radians 3 00:00:20,340 --> 00:00:21,150 ‫per second. 4 00:00:21,540 --> 00:00:32,250 ‫So this would be P here, this would be Q and this would be R, so the resultant vector from the origin 5 00:00:32,910 --> 00:00:34,190 ‫will be something like this. 6 00:00:34,800 --> 00:00:45,540 ‫Now by inspection, you know that the coordinates of this vector in the inertial frame are one one and 7 00:00:45,540 --> 00:00:47,100 ‫then minus one. 8 00:00:47,700 --> 00:00:55,530 ‫With respect to the inertia frame, you have a component in the positive X direction, positive Y direction 9 00:00:55,830 --> 00:00:59,610 ‫and then negative Z direction in the body frame. 10 00:00:59,610 --> 00:01:03,990 ‫It would still be one, one and one. 11 00:01:04,590 --> 00:01:07,770 ‫But let's use our rotation matrix approach. 12 00:01:08,340 --> 00:01:12,780 ‫In the last video I showed you this matrix r z. 13 00:01:13,260 --> 00:01:23,460 ‫Since we rotated power to Radians about the inertial Z axis, that means that our BPCI equals PI over 14 00:01:23,460 --> 00:01:24,390 ‫to radians. 15 00:01:24,720 --> 00:01:35,070 ‫So if you substitute BPCI with PI over two, then you will get this matrix here to calculate our coordinates 16 00:01:35,070 --> 00:01:43,770 ‫in the inertia frame we write X, Y, Z, which are our coordinates in the inertia frame equals R, 17 00:01:43,770 --> 00:01:51,990 ‫Z times are Y times R, X, and then we multiply this product of matrices by the coordinates in the 18 00:01:51,990 --> 00:01:56,190 ‫body frame, in this body frame here in red. 19 00:01:56,700 --> 00:02:01,740 ‫From the last video, you know what this product is. 20 00:02:02,250 --> 00:02:03,810 ‫It's this one here. 21 00:02:04,350 --> 00:02:07,470 ‫And so we take our R Z matrix. 22 00:02:08,130 --> 00:02:18,840 ‫We multiply it by this matrix, which is our R Y times are X times the coordinate values in the body 23 00:02:18,840 --> 00:02:19,320 ‫frame. 24 00:02:19,800 --> 00:02:28,050 ‫And so if you multiply them all together, then you will get our Z times are Y times R X, which is 25 00:02:28,530 --> 00:02:29,820 ‫this one here. 26 00:02:30,360 --> 00:02:40,250 ‫And then you multiply that by this vector and you will end up with one one and minus one. 27 00:02:40,830 --> 00:02:48,460 ‫And as you can see, X equals one, Y equals one, and then Z equals minus one. 28 00:02:49,140 --> 00:02:56,070 ‫These are the inertia frame coordinates and you can also see it here by inspection. 29 00:02:56,640 --> 00:02:59,610 ‫And that's the purpose of rotation matrices. 30 00:03:00,120 --> 00:03:08,190 ‫Let's say that, you know, the coordinates of a vector in the body frame, which is our case here. 31 00:03:08,640 --> 00:03:17,160 ‫However, you would like to know the coordinates of the same vector, but in the initial frame, this 32 00:03:17,160 --> 00:03:21,510 ‫vector, the purple one, it's not changing. 33 00:03:21,870 --> 00:03:22,920 ‫It is what it is. 34 00:03:23,430 --> 00:03:31,320 ‫But if you first measured that vector in the body frame, then thanks to rotation matrices, you can 35 00:03:31,320 --> 00:03:41,160 ‫instantly compute the coordinates of the same vector, but in a different frame in the inertia frame. 36 00:03:41,670 --> 00:03:52,230 ‫And of course, if you're Phi Theta and PSI, if they're all pi over to radians, then you don't really 37 00:03:52,230 --> 00:03:53,940 ‫need rotation matrices. 38 00:03:54,420 --> 00:04:05,310 ‫But what if your, let's say PHI is five or six radians, then Theta is PI over three radians and then 39 00:04:05,330 --> 00:04:11,430 ‫BPCI is minus Pi over four radians. 40 00:04:11,970 --> 00:04:14,100 ‫Then you cannot visually inspected. 41 00:04:14,100 --> 00:04:21,930 ‫Right, but you can put your angles into the rotation matrices and then find the coordinates of the 42 00:04:21,930 --> 00:04:34,230 ‫vector in the inertia frame like this x, y, z equals then our Z where psi equals minus pi over four 43 00:04:34,230 --> 00:04:46,530 ‫radians times are Y where theta equals pi over three irradiance times are X, where phi equals five 44 00:04:46,530 --> 00:04:55,560 ‫or six radians times your angular velocity vector in the body frame you put your angles in, you will 45 00:04:55,560 --> 00:05:00,210 ‫have a three by three matrix here, which will be a product of. 46 00:05:00,580 --> 00:05:09,130 ‫These three matrices and then the vector that you had initially measured in the body frame, once you 47 00:05:09,130 --> 00:05:17,140 ‫multiply this product of matrices by the body frame coordinates, you will know the coordinates of that 48 00:05:17,140 --> 00:05:18,770 ‫vector in the initial frame. 49 00:05:19,240 --> 00:05:26,050 ‫So that's why they are used in engineering and that's why we will be using them in this course as well. 50 00:05:26,680 --> 00:05:27,610 ‫Thank you very much.