WEBVTT 00:00:00.350 --> 00:00:02.260 - [Instructor] We're told a boat is traveling 00:00:02.260 --> 00:00:05.900 at a speed of 26 kilometers per hour in a direction 00:00:05.900 --> 00:00:09.940 that is a 300 degree rotation from East. 00:00:09.940 --> 00:00:12.630 At a certain point it encounters a current 00:00:12.630 --> 00:00:16.070 at a speed of 15 kilometers per hour in a direction 00:00:16.070 --> 00:00:19.720 that is a 25 degree rotation from East. 00:00:19.720 --> 00:00:22.380 Answer two questions about the boat's velocity 00:00:22.380 --> 00:00:24.470 after it meets the current. 00:00:24.470 --> 00:00:25.850 Alright, the first question is, 00:00:25.850 --> 00:00:29.110 what is the boat's speed after it meets the current? 00:00:29.110 --> 00:00:31.390 And it says, round your answer to the nearest 10th. 00:00:31.390 --> 00:00:34.140 You can round intermediate values to the nearest 100th. 00:00:34.140 --> 00:00:37.070 And what is the direction of the boat's velocity 00:00:37.070 --> 00:00:38.670 after it meets the current? 00:00:38.670 --> 00:00:40.730 And they say the same, well, they actually here it say, 00:00:40.730 --> 00:00:42.500 round your answer to the nearest integer 00:00:42.500 --> 00:00:45.570 and you can round intermediate values to the nearest 100th. 00:00:45.570 --> 00:00:47.170 So like always pause this video 00:00:47.170 --> 00:00:49.020 and see if you can work through this. 00:00:50.070 --> 00:00:52.120 All right, now let's work on this together. 00:00:52.120 --> 00:00:54.230 So first let's visualize each of these vectors. 00:00:54.230 --> 00:00:57.610 We have this vector 26 kilometers per hour in a direction 00:00:57.610 --> 00:01:00.270 that is a 300 degree rotation from East. 00:01:00.270 --> 00:01:04.110 And we have this vector 15 kilometers per hour 00:01:04.110 --> 00:01:08.210 in a direction that is a 25 degree rotation from East. 00:01:08.210 --> 00:01:12.120 And so let me draw some axes here. 00:01:12.120 --> 00:01:16.990 So let's say that is my Y-axis. 00:01:16.990 --> 00:01:21.990 And then let's say that this over here is my X-axis. 00:01:22.000 --> 00:01:26.140 And then that first vector 300 degree rotation from East, 00:01:26.140 --> 00:01:28.660 East is in the positive X direction. 00:01:28.660 --> 00:01:33.170 This would be 90 degrees, 180 degrees, 270 degrees. 00:01:33.170 --> 00:01:35.390 I'm going counter-clockwise cause that's the convention 00:01:35.390 --> 00:01:37.280 for a positive angle. 00:01:37.280 --> 00:01:40.490 And then we'd go a little bit past 270, 00:01:40.490 --> 00:01:44.810 we would go right, right over there. 00:01:44.810 --> 00:01:48.240 And the magnitude of this vector is 26 kilometers per hour. 00:01:48.240 --> 00:01:50.970 I'll just write a 26 right over there. 00:01:50.970 --> 00:01:54.230 And then this other vector which is the current 00:01:54.230 --> 00:01:56.060 15 kilometers per hour in a direction 00:01:56.060 --> 00:01:58.230 that is a 25 degree rotation from East. 00:01:58.230 --> 00:02:02.880 So 25 degree rotation might be something like this 00:02:02.880 --> 00:02:04.340 and it's going to be shorter. 00:02:04.340 --> 00:02:06.720 It's 15 kilometers per hour. 00:02:06.720 --> 00:02:10.540 So, it's going to be roughly about that long. 00:02:10.540 --> 00:02:12.780 I'm obviously just approximating it 00:02:12.780 --> 00:02:15.860 and I'll just write 15 there for its magnitude. 00:02:15.860 --> 00:02:19.410 So we can visualize what the boat's speed 00:02:19.410 --> 00:02:21.770 and direction it is after it meets the current. 00:02:21.770 --> 00:02:24.350 It's going to be the sum of these two vectors. 00:02:24.350 --> 00:02:27.480 And so if we wanted to sum these two vectors 00:02:27.480 --> 00:02:30.640 we could put the tail of one at the head of the other. 00:02:30.640 --> 00:02:33.440 And so let's shift this blue vector down here. 00:02:33.440 --> 00:02:36.260 So it's at the head of the red vector. 00:02:36.260 --> 00:02:38.310 So it would be something like this. 00:02:38.310 --> 00:02:43.010 And so our resulting speed after it meets the current 00:02:43.010 --> 00:02:44.790 would look something like this. 00:02:44.790 --> 00:02:48.360 We've seen this in many other videos so far 00:02:48.360 --> 00:02:50.410 but we don't want to just figure it out visually. 00:02:50.410 --> 00:02:52.640 We want to actually figure out its actual speed 00:02:52.640 --> 00:02:54.430 which would be the magnitude of this vector 00:02:54.430 --> 00:02:55.900 and its actual direction. 00:02:55.900 --> 00:02:57.030 So what is the angle? 00:02:57.030 --> 00:02:58.620 And we could say it as a positive angle. 00:02:58.620 --> 00:03:01.410 So what the rotation, the positive rotation 00:03:01.410 --> 00:03:05.230 from the positive X-axis or from due East. 00:03:05.230 --> 00:03:07.530 So to do that, what I'm going to do is represent each 00:03:07.530 --> 00:03:11.430 of our original vectors in terms of their components. 00:03:11.430 --> 00:03:14.060 And so this red vector up here 00:03:14.060 --> 00:03:17.370 and we've done this multiple times explaining the intuition. 00:03:17.370 --> 00:03:20.680 It's X component is going to be its magnitude 00:03:20.680 --> 00:03:23.920 26 times the cosine of this angle, 00:03:23.920 --> 00:03:26.980 cosine of 300 degrees. 00:03:26.980 --> 00:03:31.290 And it's Y component is going to be 26 times the sine 00:03:31.290 --> 00:03:33.280 of 300 degrees. 00:03:33.280 --> 00:03:35.760 If that's unfamiliar to you, I encourage you to review it 00:03:35.760 --> 00:03:37.600 in other videos where we first introduced 00:03:37.600 --> 00:03:40.030 the notion of components, it comes straight out 00:03:40.030 --> 00:03:43.930 of our unit circle definition of trig functions. 00:03:43.930 --> 00:03:47.040 And similarly, this vector right over here 00:03:47.040 --> 00:03:49.710 it's X component is going to be its magnitude times 00:03:49.710 --> 00:03:53.150 the cosine of 25 degrees. 00:03:53.150 --> 00:03:58.150 And it's Y component is going to be 15 times the sine 00:03:58.250 --> 00:04:00.740 of 25 degrees. 00:04:00.740 --> 00:04:03.070 And now when we have it expressed this way, 00:04:03.070 --> 00:04:05.870 if we want to have the resulting vector, 00:04:05.870 --> 00:04:08.490 let's call the resulting vector S 00:04:08.490 --> 00:04:10.970 for maybe the resulting speed. 00:04:10.970 --> 00:04:13.970 Its components are going to be the sum of each of these. 00:04:13.970 --> 00:04:16.120 So we can write it over here. 00:04:16.120 --> 00:04:19.860 Vector S is going to be equal to 00:04:19.860 --> 00:04:22.550 it's going to be the X component of this red vector 00:04:22.550 --> 00:04:23.970 of our original speed vector. 00:04:23.970 --> 00:04:28.880 So, 26 cosine of 300 degrees 00:04:28.880 --> 00:04:31.710 plus the X component of the current. 00:04:31.710 --> 00:04:36.710 So, 15 times cosine of 25 degrees and then the Y components. 00:04:37.340 --> 00:04:40.230 Once again, I add the corresponding Y components 00:04:40.230 --> 00:04:45.080 26 sine of 300 degrees 00:04:45.080 --> 00:04:50.080 plus 15 sine of 25 degrees. 00:04:50.290 --> 00:04:52.170 And now we could use a calculator to figure out 00:04:52.170 --> 00:04:55.170 what these are, to say what these approximately are. 00:04:55.170 --> 00:04:57.780 So first the X component, we're going to take 00:04:57.780 --> 00:05:02.513 the cosine of 300 degrees, times 26, 00:05:03.380 --> 00:05:06.400 plus I'll open a parenthesis here. 00:05:06.400 --> 00:05:11.343 We're going to take the cosine of 25 degrees, times 15, 00:05:12.280 --> 00:05:14.210 close our parentheses. 00:05:14.210 --> 00:05:19.210 And that is equal to 26.59 if I round to the nearest 100th. 00:05:20.689 --> 00:05:23.960 26.59. 00:05:23.960 --> 00:05:25.960 And now let's do the Y component. 00:05:25.960 --> 00:05:30.960 We have the sine of 300 degrees, times 26, 00:05:32.190 --> 00:05:34.960 plus I'll open parentheses, 00:05:34.960 --> 00:05:39.960 the sine of 25 degrees times 15, close parentheses, 00:05:40.480 --> 00:05:45.480 is equal to negative 16.18 to round to the nearest 100th. 00:05:45.720 --> 00:05:48.690 Negative 16.18. 00:05:48.690 --> 00:05:52.200 And let's just make sure that this makes intuitive sense. 00:05:52.200 --> 00:05:54.460 So, 26.59. 00:05:54.460 --> 00:05:56.610 So we're going to go forward in this direction 00:05:56.610 --> 00:05:59.340 26.59 on the X direction. 00:05:59.340 --> 00:06:03.340 And then we go negative 16.18 in the Y direction. 00:06:03.340 --> 00:06:06.890 So this does seem to match our intuition 00:06:06.890 --> 00:06:09.260 when we tried to look at this visually. 00:06:09.260 --> 00:06:11.650 So we now have the X and Y components 00:06:11.650 --> 00:06:13.410 of the resulting vector 00:06:13.410 --> 00:06:14.720 but that's not what they're asking for. 00:06:14.720 --> 00:06:17.500 They're asking for the speed which would be the magnitude 00:06:17.500 --> 00:06:19.670 of this vector right over here. 00:06:19.670 --> 00:06:24.430 And so I could write the magnitude of that vector 00:06:24.430 --> 00:06:26.360 which is going to be its speed. 00:06:26.360 --> 00:06:28.510 We'll just use the Pythagorean theorem here. 00:06:28.510 --> 00:06:31.750 It is going to be the square root of this squared 00:06:31.750 --> 00:06:33.620 plus this squared, because once again 00:06:33.620 --> 00:06:35.100 this forms a right triangle here. 00:06:35.100 --> 00:06:37.340 And we review this in other videos, 00:06:37.340 --> 00:06:41.457 it's going to be the square root of 26.59 squared 00:06:42.780 --> 00:06:47.780 plus negative 16.18 squared which is approximately equal to, 00:06:48.810 --> 00:06:50.706 they want us to round to the nearest 10th, 00:06:50.706 --> 00:06:55.706 26.59 squared plus. 00:06:57.070 --> 00:06:58.600 And it doesn't matter that there's a negative here 00:06:58.600 --> 00:06:59.433 cause I'm squaring it. 00:06:59.433 --> 00:07:04.370 So I'll just write 16.18 squared is equal to that. 00:07:06.770 --> 00:07:09.500 And then we want to take the square root of that. 00:07:09.500 --> 00:07:13.693 We get 31 point, if we round to the nearest 10th, 31.1. 00:07:15.670 --> 00:07:20.360 So it's approximately 31.1 and we'll write the units here, 00:07:20.360 --> 00:07:25.240 kilometers per hour is the speed the boat's speed 00:07:25.240 --> 00:07:27.340 after it meets the current. 00:07:27.340 --> 00:07:29.720 And now the second question is what is the direction 00:07:29.720 --> 00:07:33.500 of the boat's velocity after it meets the current? 00:07:33.500 --> 00:07:35.870 Well, one way to think about it is 00:07:35.870 --> 00:07:38.250 if we look at this angle right over here 00:07:38.250 --> 00:07:40.360 which would tell us the direction 00:07:40.360 --> 00:07:44.220 the tangent of that angle, theta, let me write this down. 00:07:44.220 --> 00:07:47.070 Tangent of that angle theta. 00:07:47.070 --> 00:07:48.950 We know your tangent is your change in Y 00:07:48.950 --> 00:07:50.050 over your change in X. 00:07:50.050 --> 00:07:51.660 You can even view it as the slope 00:07:51.660 --> 00:07:53.570 of this vector right over here. 00:07:53.570 --> 00:07:56.130 We know what our changes in X or Y are. 00:07:56.130 --> 00:07:58.100 Those are X and Y components. 00:07:58.100 --> 00:07:59.460 So it's going to be our change in Y 00:07:59.460 --> 00:08:04.063 which is negative 16.18, over 26.59, 00:08:06.870 --> 00:08:08.320 our change in X. 00:08:08.320 --> 00:08:10.520 And so to solve for theta, we could say 00:08:10.520 --> 00:08:15.320 that theta will be equal to the inverse tangent. 00:08:15.320 --> 00:08:16.930 And we'll have to think about this for a second 00:08:16.930 --> 00:08:19.650 because this might not get us the exact theta that we want 00:08:19.650 --> 00:08:21.160 because the inverse tangent function 00:08:21.160 --> 00:08:24.030 is going to give us something between positive 90 degrees 00:08:24.030 --> 00:08:25.830 and negative 90 degrees. 00:08:25.830 --> 00:08:27.740 But the number we want, actually it looks 00:08:27.740 --> 00:08:31.811 like it's going to be between 270 and 360 degrees 00:08:31.811 --> 00:08:33.650 because we're doing a, 00:08:33.650 --> 00:08:35.630 we want to think about a positive rotation 00:08:35.630 --> 00:08:38.380 instead of a negative one but let's just try to evaluate it. 00:08:38.380 --> 00:08:40.530 The inverse tan of this, 00:08:40.530 --> 00:08:45.073 of negative 16.18, over 26.59. 00:08:47.284 --> 00:08:50.010 16.18 negative 00:08:50.910 --> 00:08:55.910 divided by 26.59 is equal to this. 00:08:57.100 --> 00:09:00.840 And now I am going to take the inverse tangent of that. 00:09:00.840 --> 00:09:05.690 And that gets us negative 31 degrees, which makes sense. 00:09:05.690 --> 00:09:06.740 This looks intuitive sense 00:09:06.740 --> 00:09:09.230 that if you were to do a clockwise rotation 00:09:09.230 --> 00:09:11.660 which would be a negative angle from the positive X-axis 00:09:11.660 --> 00:09:14.290 it looks like what we drew, but let's just go 00:09:14.290 --> 00:09:15.880 with the convention of everything else here. 00:09:15.880 --> 00:09:17.680 And let's try to have a positive angle. 00:09:17.680 --> 00:09:21.070 So what we can do is add 360 degrees to that 00:09:21.070 --> 00:09:22.570 to make a full rotation around. 00:09:22.570 --> 00:09:24.710 And we essentially have the equivalent angle. 00:09:24.710 --> 00:09:29.480 So let's add 360 to that to get that right over there. 00:09:29.480 --> 00:09:31.170 So if we round to the nearest integer 00:09:31.170 --> 00:09:35.840 we're looking at approximately 329 degrees. 00:09:35.840 --> 00:09:40.840 So theta is approximately 329 degrees. 00:09:41.110 --> 00:09:43.680 So here, when I said theta is equal to this I could write 00:09:43.680 --> 00:09:48.680 theta is going to be equal to this plus 360 degrees. 00:09:49.140 --> 00:09:52.110 Now what's interesting is, I was able to add 360 degrees 00:09:52.110 --> 00:09:54.120 to get to the exact same place. 00:09:54.120 --> 00:09:57.350 If we had a situation where our angle was actually 00:09:57.350 --> 00:09:59.310 this angle right over here not the situation 00:09:59.310 --> 00:10:00.270 that we actually dealt with, 00:10:00.270 --> 00:10:02.170 but if it was in the second quadrant, 00:10:02.170 --> 00:10:04.070 we would have gotten this theta. 00:10:04.070 --> 00:10:06.510 And we would have had to be able to realize that, 00:10:06.510 --> 00:10:08.030 hey we're dealing with the second quadrant 00:10:08.030 --> 00:10:09.420 that has the same slope. 00:10:09.420 --> 00:10:11.080 So instead of adding 360 degrees 00:10:11.080 --> 00:10:12.950 we would have added 180 degrees. 00:10:12.950 --> 00:10:16.153 And we've also covered that in other videos as well.