WEBVTT 00:00:00.000 --> 00:00:03.160 This is Giancoli Answers with Mr. Dychko. 00:00:04.240 --> 00:00:06.620 The way I like to think about subtracting vectors is 00:00:06.620 --> 00:00:08.080 adding the opposite 00:00:08.280 --> 00:00:11.220 because then you can use the head to tail method of adding vectors. 00:00:11.340 --> 00:00:14.420 So we have vector A which is redrawn here 00:00:14.620 --> 00:00:17.760 and on the head of vector A, you put the tail of 00:00:17.760 --> 00:00:21.780 the opposite of C so the opposite of C is this vector 00:00:21.780 --> 00:00:24.140 in the other direction, straight up in this case 00:00:24.380 --> 00:00:27.140 so we have the green one shown here. 00:00:28.880 --> 00:00:33.020 And then the resultant goes from the very beginning to the very end 00:00:33.020 --> 00:00:35.120 so that's drawn in red here. 00:00:35.840 --> 00:00:41.500 So we'll find the length of the resultant by finding its components first 00:00:41.500 --> 00:00:43.240 and then use Pythagoras and then 00:00:43.240 --> 00:00:45.220 use inverse tangent to get its angle 00:00:45.220 --> 00:00:47.960 and that's the usual routine for vectors. 00:00:48.160 --> 00:00:50.780 So the resultant x-component is 00:00:50.920 --> 00:00:53.140 the x-component of vector A because 00:00:53.140 --> 00:00:55.720 vector negative C has no x-component 00:00:55.720 --> 00:00:58.960 so we have 44 times cos 28 00:00:59.600 --> 00:01:01.940 so length of vector A times cos of its angle here 00:01:01.940 --> 00:01:06.100 gives the x-component of the resultant 00:01:08.760 --> 00:01:11.300 and that's 38.8497 00:01:11.440 --> 00:01:14.020 and then the y-component of the resultant will be 00:01:14.020 --> 00:01:16.380 the y-component of A 00:01:16.860 --> 00:01:18.240 that's this part here 00:01:18.240 --> 00:01:23.060 and that's found by taking the hypotenuse or length of A—44— 00:01:23.060 --> 00:01:25.400 and multiply by sin of 28; 00:01:25.400 --> 00:01:29.260 sin is the opposite divided by hypotenuse so when you multiply by the hypotenuse 00:01:29.260 --> 00:01:32.040 and then times by sin 28, you get the opposite 00:01:32.300 --> 00:01:37.560 and then add to that 31 and we get 51.657; 00:01:38.660 --> 00:01:41.700 I don't know, just in case, it's helpful to explain that a bit better 00:01:41.700 --> 00:01:46.140 we have sin 28, by definition, is opposite 00:01:46.140 --> 00:01:49.980 which in this case is y-component of A divided by hypotenuse which is A 00:01:49.980 --> 00:01:53.800 and then multiply both sides by A here 00:01:55.180 --> 00:01:57.780 and then A's cancel and then switch the sides around, 00:01:57.780 --> 00:02:00.740 you get A y is A times sin 28 00:02:01.380 --> 00:02:02.940 and that's what we have put here 00:02:02.940 --> 00:02:05.520 and then we get 51.657 there 00:02:05.520 --> 00:02:11.020 and their resultant magnitude is the square root of the sum of the squares 00:02:11.160 --> 00:02:13.300 so square both of these components 00:02:13.300 --> 00:02:17.360 add them, take the square root and you get 64.6 units 00:02:17.360 --> 00:02:20.220 and then the angle Θ for the resultant is 00:02:20.220 --> 00:02:25.440 the opposite which is this length here— 00:02:25.740 --> 00:02:27.720 oops, it's going upwards though— 00:02:28.400 --> 00:02:32.560 so that's resultant y-direction 00:02:32.720 --> 00:02:39.920 that length divided by its x-component and so we get 00:02:40.820 --> 00:02:44.080 inverse tangent of 51.657 divided by 00:02:45.920 --> 00:02:50.000 divided by the x-component of 38.8497 00:02:50.140 --> 00:02:52.460 and that gives 53.1 degrees 00:02:52.460 --> 00:02:56.740 and from the drawing, we can see that that's above the positive x-axis.