WEBVTT
00:00:03.140 --> 00:00:10.760
We already know that a vector is a set of numbers that can be represented in a suitable space by a line segment with a specific length and direction.
00:00:11.310 --> 00:00:19.890
Weβve also seen that a line segment has a magnitude and direction, which basically means that we can describe it by saying how long it is and in which direction itβs pointing.
00:00:20.780 --> 00:00:30.140
In this video, weβre gonna talk about horizontal and vertical components of two-dimensional vectors and introduce the π and π unit vector notation.
00:00:33.570 --> 00:00:36.520
Any two-dimensional vector has two components.
00:00:36.830 --> 00:00:42.800
The first represents the amount of movement in the π₯-direction and the second, the amount of movement in the π¦-direction.
00:00:43.620 --> 00:00:51.140
Of course, when I say movement, Iβm just talking about differences in π₯- and π¦-coordinates in a graphical representation of the vector by a line segment on a graph.
00:00:51.670 --> 00:00:57.000
The vector itself might be representing something completely different, force or acceleration, for example.
00:00:58.620 --> 00:01:03.920
So in this case, the amount of π₯-movement is this distance here.
00:01:03.920 --> 00:01:09.610
So weβre going from an π₯-coordinate of three to an π₯-coordinate of six, so thatβs a difference of plus three.
00:01:10.080 --> 00:01:13.600
So our π₯-component of the vector is positive three.
00:01:14.280 --> 00:01:17.290
The π¦-component is this bit here.
00:01:18.980 --> 00:01:25.630
The π¦-coordinate of π΄ was two, the π¦-coordinate of π΅ was nine, so thatβs a difference of plus seven.
00:01:27.240 --> 00:01:33.220
And we can write that as π΄π΅, vector π΄π΅, in this format here: three, seven.
00:01:33.220 --> 00:01:38.910
So the three is the π₯-component and the seven is the π¦-component.
00:01:41.330 --> 00:01:49.330
And if we had a three-dimensional coordinate system with π₯- and π¦- and π§-coordinates, we would just be inserting another number at the end here onto our vector.
00:01:49.330 --> 00:01:53.460
So we can extend this system to any number of dimensions.
00:01:55.490 --> 00:02:03.120
We define two special vectors, π and π, to be positive one in the π₯-direction or positive one in the π¦-direction, respectively.
00:02:03.460 --> 00:02:07.550
So this is π, itβs just a movement of one in the π₯-direction.
00:02:07.930 --> 00:02:11.740
And hereβs π, a movement of one in the π¦-direction.
00:02:13.040 --> 00:02:20.180
So remember that the π and π vectors β are at one, zero and zero, one β can be placed anywhere on the graph.
00:02:20.180 --> 00:02:21.860
They donβt have to start at the origin.
00:02:23.130 --> 00:02:24.720
So there we are; I placed them somewhere else.
00:02:25.090 --> 00:02:28.590
Each vector is just describing a particular journey.
00:02:28.640 --> 00:02:35.280
In this particular case here, for π, weβre adding one to the π₯-coordinate and weβre leaving the π¦-coordinate as it is.
00:02:35.280 --> 00:02:37.590
So weβre doing this journey from here to here.
00:02:38.640 --> 00:02:43.290
In the π case, weβre not adding anything to the π₯-coordinate, but weβre adding one to the π¦-coordinate.
00:02:43.290 --> 00:02:47.550
It describes this movement from here up to here.
00:02:50.040 --> 00:02:56.930
Now remember, we can also use the rules of adding and subtracting vectors to stack up these πs and πs, to create bigger vectors.
00:02:57.240 --> 00:03:03.740
So, for example, this vector here, π, represents a movement of one in the π₯-direction and none in the π¦-direction.
00:03:03.960 --> 00:03:13.330
If I increase that to this vector here, so this whole length here, that would be two πs following on from each other.
00:03:13.630 --> 00:03:19.020
Or, it will be a translation of two in the π₯-direction and zero in the π¦-direction.
00:03:20.390 --> 00:03:38.550
Now if I added on to that this vector here β which starts at the end here and then moves up not one, not two, but three β that would be three π vectors added on together, making a vector of three π or zero, three.
00:03:38.890 --> 00:03:48.420
So if I add two π plus three π, that represents this green journey from the start point to the end point up here.
00:03:48.890 --> 00:03:52.060
So two π, the π₯-component would be two.
00:03:52.410 --> 00:03:55.190
Three π, the π¦-component would be three.
00:03:55.490 --> 00:03:59.450
And the direct journey two π plus three π is the green line.
00:03:59.450 --> 00:04:04.060
So this means the π₯-component was two and the π¦-component was three.
00:04:04.460 --> 00:04:09.240
So weβve now got two different ways of representing this green vector here.
00:04:09.240 --> 00:04:12.440
We can use it in the standard vector notation that weβre familiar with.
00:04:12.640 --> 00:04:16.960
But weβve got this new notation here, in terms of π and π vectors.
00:04:16.960 --> 00:04:23.670
The number of steps in the π₯-direction is the πs and the number of steps in the π¦-direction is the πs.
00:04:25.460 --> 00:04:27.510
So letβs sum that up in the general case.
00:04:27.540 --> 00:04:50.490
If we start off at point πΆ here with coordinates π₯ one, π¦ one and we end up at point π· here with coordinates π₯ two, π¦ two, then vector πΆπ· is, the π₯-component here is the difference in the π₯-coordinates and the π¦-component here is the difference in the π¦-coordinates.
00:04:51.880 --> 00:04:54.010
So thatβs our standard vector way of writing it.
00:04:54.260 --> 00:05:07.630
Or, we can say that that is π₯ two minus π₯ one lots of the π unit vector plus π¦ two minus π¦ one lots of the π unit vector.
00:05:10.000 --> 00:05:11.210
That make some kind of sense.
00:05:11.210 --> 00:05:22.270
All weβre saying is that the π₯-component of a vector is the coefficient of π, in this format, and the π¦-component of the vector is the coefficient of π, in this format.
00:05:22.640 --> 00:05:35.790
So weβve just got this new notation where we have the π vector, which is a step of one in the π₯-direction, the π vector, which is a step of one in the π¦-direction, and we just say how many of those weβre doing in each case to make up our vector.
00:05:37.540 --> 00:05:37.840
Right.
00:05:37.840 --> 00:05:39.230
So now we know this new format.
00:05:39.230 --> 00:05:43.870
Letβs just have a look at a couple of quick questions that involve using πs and πs in our questions.
00:05:44.190 --> 00:05:46.010
So weβre going to do this question here.
00:05:46.010 --> 00:05:49.020
Weβve got π΄π΅; itβs the vector three π plus four π.
00:05:49.440 --> 00:05:52.680
ππ is the vector negative two π plus three π.
00:05:52.870 --> 00:05:55.720
And weβve just got to add those two vectors together.
00:05:57.260 --> 00:06:03.150
So the first stage is just to take vector π΄π΅ and add vector ππ.
00:06:04.740 --> 00:06:09.850
And all we have to do is add the πs together first, and then add the πs together second.
00:06:11.390 --> 00:06:17.120
So three π add negative two π is just one π, so thatβs π.
00:06:19.310 --> 00:06:23.750
And four π add positive three π is seven π.
00:06:27.940 --> 00:06:30.320
So thereβs our answer, π plus seven π.
00:06:30.400 --> 00:06:41.370
Simple as that, add the π-components together, add the π-components together, look out for the negative signs and- when youβre doing those calculations; but otherwise, thatβs a pretty straightforward process.
00:06:42.520 --> 00:06:44.140
So letβs just visualise that example.
00:06:44.140 --> 00:06:46.270
So we had three π plus four π.
00:06:46.270 --> 00:06:53.790
So weβll be going positive three π and then positive four π, this is the ππ vector.
00:06:55.280 --> 00:06:56.050
So there we are.
00:06:56.050 --> 00:06:57.600
Weβve just laid that down at the origin.
00:06:57.600 --> 00:07:00.120
We couldβve laid it anywhere on the-on the graph.
00:07:01.740 --> 00:07:04.240
Now adding vectors, we just lay them end-to-end.
00:07:04.240 --> 00:07:07.750
So what we were adding, π₯π¦, was negative two π plus three π.
00:07:08.120 --> 00:07:15.950
So weβre effectually starting π₯ from π΅, so π₯ lays on top of π΅ and weβre going negative two π.
00:07:15.950 --> 00:07:25.510
So weβre going two in the negative π₯-direction and weβre going positive three π, up to here.
00:07:27.660 --> 00:07:31.170
So adding vectors is just a matter of laying them end-to-end on the graph.
00:07:31.170 --> 00:07:38.780
So we laid π΄π΅ down, which started here and ended here, and then we just added ππ to the end of that, laid that onto the end.
00:07:38.780 --> 00:07:42.290
So that started from where we just finished off and then ended up here.
00:07:43.700 --> 00:07:46.180
So the resultant vector is this green one here.
00:07:46.500 --> 00:07:53.710
And to get from the beginning of the green vector to the end of the green vector, we had to go positive one in the π₯-direction.
00:07:53.710 --> 00:07:56.190
So thatβs one π, or just π.
00:07:56.990 --> 00:08:01.650
And in the π¦-direction, weβre going up seven all the way up here.
00:08:02.330 --> 00:08:05.820
So thatβs plus seven π.
00:08:08.990 --> 00:08:15.660
So when youβre doing these questions, it really is just a matter of adding the π₯-components together, adding the π¦-components together, and coming up with a simple answer.
00:08:15.660 --> 00:08:17.870
You donβt need to do all this graphical checking.
00:08:17.870 --> 00:08:22.790
But Iβm just hoping that thatβs giving you some extra insight into the process and why it works.
00:08:24.230 --> 00:08:24.530
Right.
00:08:24.530 --> 00:08:26.360
Letβs take a look at one final question then.
00:08:26.590 --> 00:08:30.480
Weβve got vector π΄π΅, is three π take away four π.
00:08:30.950 --> 00:08:34.870
Vector ππ is negative four π add seven π.
00:08:35.030 --> 00:08:39.370
And weβve got to find vector π΄π΅ take away vector ππ.
00:08:40.660 --> 00:08:45.510
So just writing that out, π΄π΅ is three π take away four π.
00:08:45.840 --> 00:08:51.380
ππ is negative four π plus seven π, and thatβs what weβre taking away from vector π΄π΅.
00:08:51.910 --> 00:08:59.460
So we just need to be really careful, when weβre taking these away, about the signs here because weβve got a negative outside the bracket, weβve got negatives and positive inside the bracket.
00:09:01.620 --> 00:09:03.910
So letβs just start off with the π-components then.
00:09:04.140 --> 00:09:09.320
Iβve got three π and Iβm taking away negative four π, so that means Iβm adding four π.
00:09:09.320 --> 00:09:11.850
So three π add four π is seven π.
00:09:13.120 --> 00:09:26.820
And then for the π-components, Iβm starting off with negative four π and Iβm taking away positive seven π, so f- negative four take away another seven is negative eleven π.
00:09:29.160 --> 00:09:32.510
So thereβs our answer, seven π take away eleven π.
00:09:33.960 --> 00:09:36.050
And just to summarise then what weβve learnt.
00:09:36.390 --> 00:09:44.810
π is the unit vector in the π₯-direction one, zero and π is the unit vector in the π¦-direction zero, one.
00:09:46.400 --> 00:09:53.580
And given any vector β like π΄π΅ is five, three β this here is the π₯-component, this here is the π¦-component.
00:09:53.580 --> 00:10:02.260
We can rewrite this as five π, because thatβs the number of π₯s, plus three π, because thatβs the number of π¦s.
00:10:02.260 --> 00:10:17.200
And given two vectors, π΄π΅ and πΆπ·, with their π- and π-components like this, in this format, we can add or subtract them just by adding or subtracting their π-components and their π-components, separately.
00:10:18.290 --> 00:10:30.770
So, for example, adding π΄π΅ and πΆπ·, we can add the two and the negative two π and we can add the negative three and the four π.
00:10:32.340 --> 00:10:38.270
Well, in this case, two plus negative two π, thatβs zero πs, so we donβt need to bother writing zero π.
00:10:38.600 --> 00:10:44.590
And negative three plus four is just positive one, so we end up with an answer of just π, one π.
00:10:46.710 --> 00:11:02.760
And if we want to subtract the vectors π΄π΅ minus πΆπ·, weβve got two of the πs here take away negative two πs, and we had negative three take away four of the πs.
00:11:05.160 --> 00:11:10.730
Then two take away negative two is two add two, so thatβs four of the πs.
00:11:11.390 --> 00:11:17.580
And negative three take away another four is negative seven πs.
00:11:18.290 --> 00:11:20.290
So weβre adding negative seven π.
00:11:20.680 --> 00:11:29.400
So we probably not write plus negative π, we just write four π take away seven π.
00:11:32.410 --> 00:11:40.740
So hopefully, youβll be comfortable using the π and π unit vectors just to represent the π₯- and the π¦-component of any vectors that you come across now.