0:00:00.630,0:00:02.090 - [Instructor] In other[br]videos, we have talked 0:00:02.090,0:00:04.100 about how a vector can[br]be completely defined 0:00:04.100,0:00:06.880 by a magnitude and a[br]direction, you need both. 0:00:06.880,0:00:08.260 And here we have done that. 0:00:08.260,0:00:09.880 We have said that the magnitude 0:00:09.880,0:00:12.570 of vector a is equal to three units, 0:00:12.570,0:00:15.190 these parallel lines here on both sides, 0:00:15.190,0:00:17.170 it looks like a double absolute value. 0:00:17.170,0:00:19.090 That means the magnitude of vector a. 0:00:19.090,0:00:23.150 And you can also specify[br]that visually by making sure 0:00:23.150,0:00:26.200 that the length of this vector[br]arrow is three units long. 0:00:26.200,0:00:27.560 And we also have its direction. 0:00:27.560,0:00:29.610 We see the direction of[br]vector a is 30 degrees 0:00:29.610,0:00:32.270 counter-clockwise of due East. 0:00:32.270,0:00:34.860 Now in this video, we're[br]gonna talk about other ways 0:00:34.860,0:00:38.220 or another way to specify[br]or to define a vector. 0:00:38.220,0:00:41.050 And that's by using components. 0:00:41.050,0:00:42.530 And the way that we're gonna do it is, 0:00:42.530,0:00:44.100 we're gonna think about the tail 0:00:44.100,0:00:47.300 of this vector and the[br]head of this vector. 0:00:47.300,0:00:50.450 And think about as we go[br]from the tail to the head, 0:00:50.450,0:00:53.990 what is our change in x? 0:00:53.990,0:00:55.180 And we could see our change 0:00:55.180,0:00:58.340 in x would be that right over there. 0:00:58.340,0:01:00.980 We're going from this x[br]value to this x value. 0:01:00.980,0:01:05.370 And then what is going[br]to be our change in y. 0:01:05.370,0:01:07.980 And if we're going from[br]down here to up here, 0:01:07.980,0:01:12.310 our change in y, we can[br]also specify like that. 0:01:12.310,0:01:13.500 So let me label these. 0:01:13.500,0:01:18.500 This is my change in x, and[br]then this is my change in y. 0:01:19.060,0:01:19.920 And if you think about it, 0:01:19.920,0:01:22.780 if someone told you your[br]change in x and change in y, 0:01:22.780,0:01:25.390 you could reconstruct this[br]vector right over here 0:01:25.390,0:01:27.490 by starting here, having that change in x, 0:01:27.490,0:01:31.200 then having the change in y[br]and then defining where the tip 0:01:31.200,0:01:34.740 of the vector would be[br]relative to the tail. 0:01:34.740,0:01:38.800 The notation for this is[br]we would say that vector a 0:01:38.800,0:01:42.870 is equal to, and we'll have parenthesis, 0:01:42.870,0:01:46.290 and we'll have our change[br]in x comma, change in y. 0:01:46.290,0:01:47.780 And so if we wanted to get tangible 0:01:47.780,0:01:50.340 for this particular[br]vector right over here, 0:01:50.340,0:01:53.550 we know the length of[br]this vector is three. 0:01:53.550,0:01:55.540 Its magnitude is three. 0:01:55.540,0:01:58.350 We know that this is, since[br]this is going due horizontally 0:01:58.350,0:02:00.290 and then this is going[br]straight up and down. 0:02:00.290,0:02:02.420 This is a right triangle. 0:02:02.420,0:02:05.170 And so we can use a little[br]bit of geometry from the past. 0:02:05.170,0:02:08.020 Don't worry if you need a little[br]bit of a refresher on this, 0:02:08.020,0:02:09.620 but we could use a little bit of geometry, 0:02:09.620,0:02:11.490 or a little bit of[br]trigonometry to establish, 0:02:11.490,0:02:13.610 if we know this angle,[br]if we know the length 0:02:13.610,0:02:17.210 of this hypotenuse, that[br]this side that's opposite 0:02:17.210,0:02:20.180 the 30 degree angle is gonna[br]be half the hypotenuse, 0:02:20.180,0:02:22.020 so it's going to be 3/2. 0:02:22.020,0:02:24.200 And that the change in x is going to be 0:02:24.200,0:02:26.960 the square root of three times the 3/2. 0:02:26.960,0:02:31.080 So it's going to be three,[br]square roots of three over two. 0:02:31.080,0:02:33.980 And so up here, we would[br]write our x component 0:02:33.980,0:02:37.680 is three times the square[br]root of three over two. 0:02:37.680,0:02:42.420 And we would write that[br]the y component is 3/2. 0:02:42.420,0:02:43.820 Now I know a lot of you might be thinking 0:02:43.820,0:02:47.260 this looks a lot like coordinates[br]in the coordinate plane, 0:02:47.260,0:02:48.580 where this would be the x coordinate 0:02:48.580,0:02:50.300 and this would be the y coordinate. 0:02:50.300,0:02:51.970 But when you're dealing with vectors, 0:02:51.970,0:02:54.610 that's not exactly the interpretation. 0:02:54.610,0:02:57.000 It is the case that if the vector's tail 0:02:57.000,0:03:00.860 were at the origin right[br]over here, then its head 0:03:00.860,0:03:04.670 would be at these coordinates[br]on the coordinate plane. 0:03:04.670,0:03:07.470 But we know that a vector is not defined 0:03:07.470,0:03:10.180 by its position, by the[br]position of the tail. 0:03:10.180,0:03:12.200 I could shift this vector around wherever 0:03:12.200,0:03:13.840 and it would still be the same vector. 0:03:13.840,0:03:15.590 It can start wherever. 0:03:15.590,0:03:19.000 So when you use this[br]notation in a vector context, 0:03:19.000,0:03:21.440 these aren't x coordinates[br]and y coordinates. 0:03:21.440,0:03:26.440 This is our change in x,[br]and this is our change in y. 0:03:27.070,0:03:28.480 Let me do one more example to show 0:03:28.480,0:03:30.880 that we can actually go the other way. 0:03:30.880,0:03:34.790 So let's say I defined some vector b, 0:03:34.790,0:03:39.200 and let's say that its x[br]component is square root of two. 0:03:39.200,0:03:43.520 And let's say that its y[br]component is square root of two. 0:03:43.520,0:03:46.260 So let's think about what[br]that vector would look like. 0:03:46.260,0:03:49.380 So it would, if this is its tail, 0:03:49.380,0:03:51.410 and its x component which is its change 0:03:51.410,0:03:53.030 in x is square root of two. 0:03:53.030,0:03:55.460 So it might look something like this. 0:03:55.460,0:04:00.460 So that would be change in x[br]is equal to square root of two. 0:04:00.800,0:04:03.980 And then its y component would[br]also be square root of two. 0:04:03.980,0:04:07.230 So I could write our change in y over here 0:04:07.230,0:04:08.970 is square root of two. 0:04:08.970,0:04:12.850 And so the vector would[br]look something like this. 0:04:12.850,0:04:17.850 It would start here and[br]then it would go over here, 0:04:18.580,0:04:20.590 and we can use a little bit of geometry 0:04:20.590,0:04:21.980 to figure out the magnitude 0:04:21.980,0:04:24.260 and the direction of this vector. 0:04:24.260,0:04:26.760 You can use the Pythagorean[br]theorem to establish 0:04:26.760,0:04:28.760 that this squared plus this squared 0:04:28.760,0:04:30.410 is gonna be equal to that squared. 0:04:30.410,0:04:32.380 And if you do that,[br]you're going to get this 0:04:32.380,0:04:34.510 having a length of two, which tells you 0:04:34.510,0:04:39.370 that the magnitude of[br]vector b is equal to two. 0:04:39.370,0:04:42.420 And if you wanted to figure[br]out this angle right over here, 0:04:42.420,0:04:43.870 you could do a little bit of trigonometry 0:04:43.870,0:04:46.110 or even a little bit[br]of geometry recognizing 0:04:46.110,0:04:49.500 that this is going to be a[br]right angle right over here, 0:04:49.500,0:04:52.130 and that this side and that[br]side have the same length. 0:04:52.130,0:04:53.410 So these are gonna be the same angles 0:04:53.410,0:04:55.600 which are gonna be 45 degree angles. 0:04:55.600,0:04:58.690 And so just like that, you could[br]also specify the direction, 0:04:58.690,0:05:02.770 45 degrees counter-clockwise of due East. 0:05:02.770,0:05:05.360 So hopefully you appreciate[br]that these are equivalent ways 0:05:05.360,0:05:06.540 of representing a vector. 0:05:06.540,0:05:08.950 You either can have a[br]magnitude and a direction, 0:05:08.950,0:05:10.200 or you can have your components 0:05:10.200,0:05:12.350 and you can go back and[br]forth between the two. 0:05:12.350,0:05:15.283 And we'll get more practice[br]of that in future videos.