Video Transcript
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. We’ve 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 it’s pointing. In this video, we’re gonna talk
about horizontal and vertical components of two-dimensional vectors and introduce
the 𝑖 and 𝑗 unit vector notation.
Any two-dimensional vector has two
components. The first represents the amount of
movement in the 𝑥-direction and the second, the amount of movement in the
𝑦-direction. Of course, when I say movement, I’m
just talking about differences in 𝑥- and 𝑦-coordinates in a graphical
representation of the vector by a line segment on a graph. The vector itself might be
representing something completely different, force or acceleration, for example. So in this case, the amount of
𝑥-movement is this distance here. So we’re going from an
𝑥-coordinate of three to an 𝑥-coordinate of six, so that’s a difference of plus
three. So our 𝑥-component of the vector
is positive three. The 𝑦-component is this bit
here. The 𝑦-coordinate of 𝐴 was two,
the 𝑦-coordinate of 𝐵 was nine, so that’s a difference of plus seven. And we can write that as 𝐴𝐵,
vector 𝐴𝐵, in this format here: three, seven. So the three is the 𝑥-component
and the seven is the 𝑦-component.
And if we had a three-dimensional
coordinate system with 𝑥- and 𝑦- and 𝑧-coordinates, we would just be inserting
another number at the end here onto our vector. So we can extend this system to any
number of dimensions.
We define two special vectors, 𝑖
and 𝑗, to be positive one in the 𝑥-direction or positive one in the 𝑦-direction,
respectively. So this is 𝑖, it’s just a movement
of one in the 𝑥-direction. And here’s 𝑗, a movement of one in
the 𝑦-direction. So remember that the 𝑖 and 𝑗
vectors — are at one, zero and zero, one — can be placed anywhere on the graph. They don’t have to start at the
origin. So there we are; I placed them
somewhere else. Each vector is just describing a
particular journey. In this particular case here, for
𝑖, we’re adding one to the 𝑥-coordinate and we’re leaving the 𝑦-coordinate as it
is. So we’re doing this journey from
here to here. In the 𝑗 case, we’re not adding
anything to the 𝑥-coordinate, but we’re adding one to the 𝑦-coordinate. It describes this movement from
here up to here.
Now remember, we can also use the
rules of adding and subtracting vectors to stack up these 𝑖s and 𝑗s, to create
bigger vectors. So, for example, this vector here,
𝑖, represents a movement of one in the 𝑥-direction and none in the
𝑦-direction. If I increase that to this vector
here, so this whole length here, that would be two 𝑖s following on from each
other. Or, it will be a translation of two
in the 𝑥-direction and zero in the 𝑦-direction. Now if I added on to that this
vector here — which starts at the end here and then moves up not one, not two, but
three — that would be three 𝑗 vectors added on together, making a vector of three
𝑗 or zero, three. So if I add two 𝑖 plus three 𝑗,
that represents this green journey from the start point to the end point up
here. So two 𝑖, the 𝑥-component would
be two. Three 𝑗, the 𝑦-component would be
three. And the direct journey two 𝑖 plus
three 𝑗 is the green line. So this means the 𝑥-component was
two and the 𝑦-component was three. So we’ve now got two different ways
of representing this green vector here. We can use it in the standard
vector notation that we’re familiar with. But we’ve got this new notation
here, in terms of 𝑖 and 𝑗 vectors. The number of steps in the
𝑥-direction is the 𝑖s and the number of steps in the 𝑦-direction is the 𝑗s.
So let’s sum that up in the general
case. If we start off at point 𝐶 here
with coordinates 𝑥 one, 𝑦 one and we end up at point 𝐷 here with coordinates 𝑥
two, 𝑦 two, then vector 𝐶𝐷 is, the 𝑥-component here is the difference in the
𝑥-coordinates and the 𝑦-component here is the difference in the
𝑦-coordinates. So that’s our standard vector way
of writing it. Or, we can say that that is 𝑥 two
minus 𝑥 one lots of the 𝑖 unit vector plus 𝑦 two minus 𝑦 one lots of the 𝑗 unit
vector. That make some kind of sense. All we’re saying is that the
𝑥-component of a vector is the coefficient of 𝑖, in this format, and the
𝑦-component of the vector is the coefficient of 𝑗, in this format. So we’ve just got this new notation
where we have the 𝑖 vector, which is a step of one in the 𝑥-direction, the 𝑗
vector, which is a step of one in the 𝑦-direction, and we just say how many of
those we’re doing in each case to make up our vector.
Right. So now we know this new format. Let’s just have a look at a couple
of quick questions that involve using 𝑖s and 𝑗s in our questions. So we’re going to do this question
here.
We’ve got 𝐴𝐵; it’s the vector
three 𝑖 plus four 𝑗. 𝑋𝑌 is the vector negative two 𝑖
plus three 𝑗. And we’ve just got to add those two
vectors together.
So the first stage is just to take
vector 𝐴𝐵 and add vector 𝑋𝑌. And all we have to do is add the
𝑖s together first, and then add the 𝑗s together second. So three 𝑖 add negative two 𝑖 is
just one 𝑖, so that’s 𝑖. And four 𝑗 add positive three 𝑗
is seven 𝑗. So there’s our answer, 𝑖 plus
seven 𝑗. Simple as that, add the
𝑖-components together, add the 𝑗-components together, look out for the negative
signs and- when you’re doing those calculations; but otherwise, that’s a pretty
straightforward process.
So let’s just visualise that
example. So we had three 𝑖 plus four
𝑗. So we’ll be going positive three 𝑖
and then positive four 𝑗, this is the 𝑎𝑏 vector. So there we are. We’ve just laid that down at the
origin. We could’ve laid it anywhere on
the-on the graph. Now adding vectors, we just lay
them end-to-end. So what we were adding, 𝑥𝑦, was
negative two 𝑖 plus three 𝑗. So we’re effectually starting 𝑥
from 𝐵, so 𝑥 lays on top of 𝐵 and we’re going negative two 𝑖. So we’re going two in the negative
𝑥-direction and we’re going positive three 𝑗, up to here.
So adding vectors is just a matter
of laying them end-to-end on the graph. So we laid 𝐴𝐵 down, which started
here and ended here, and then we just added 𝑋𝑌 to the end of that, laid that onto
the end. So that started from where we just
finished off and then ended up here. So the resultant vector is this
green one here. 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
𝑥-direction. So that’s one 𝑖, or just 𝑖. And in the 𝑦-direction, we’re
going up seven all the way up here. So that’s plus seven 𝑗.
So when you’re doing these
questions, it really is just a matter of adding the 𝑥-components together, adding
the 𝑦-components together, and coming up with a simple answer. You don’t need to do all this
graphical checking. But I’m just hoping that that’s
giving you some extra insight into the process and why it works.
Right. Let’s take a look at one final
question then.
We’ve got vector 𝐴𝐵, is three 𝑖
take away four 𝑗. Vector 𝑋𝑌 is negative four 𝑖 add
seven 𝑗. And we’ve got to find vector 𝐴𝐵
take away vector 𝑋𝑌.
So just writing that out, 𝐴𝐵 is
three 𝑖 take away four 𝑗. 𝑋𝑌 is negative four 𝑖 plus seven
𝑗, and that’s what we’re taking away from vector 𝐴𝐵. So we just need to be really
careful, when we’re taking these away, about the signs here because we’ve got a
negative outside the bracket, we’ve got negatives and positive inside the
bracket.
So let’s just start off with the
𝑖-components then. I’ve got three 𝑖 and I’m taking
away negative four 𝑖, so that means I’m adding four 𝑖. So three 𝑖 add four 𝑖 is seven
𝑖. And then for the 𝑗-components, I’m
starting off with negative four 𝑗 and I’m taking away positive seven 𝑗, so f-
negative four take away another seven is negative eleven 𝑗. So there’s our answer, seven 𝑖
take away eleven 𝑗.
And just to summarise then what
we’ve learnt. 𝑖 is the unit vector in the
𝑥-direction one, zero and 𝑗 is the unit vector in the 𝑦-direction zero, one. And given any vector — like 𝐴𝐵 is
five, three — this here is the 𝑥-component, this here is the 𝑦-component. We can rewrite this as five 𝑖,
because that’s the number of 𝑥s, plus three 𝑗, because that’s the number of
𝑦s. And given two vectors, 𝐴𝐵 and
𝐶𝐷, with their 𝑖- and 𝑗-components like this, in this format, we can add or
subtract them just by adding or subtracting their 𝑖-components and their
𝑗-components, separately.
So, for example, adding 𝐴𝐵 and
𝐶𝐷, we can add the two and the negative two 𝑖 and we can add the negative three
and the four 𝑗. Well, in this case, two plus
negative two 𝑖, that’s zero 𝑖s, so we don’t need to bother writing zero 𝑖. And negative three plus four is
just positive one, so we end up with an answer of just 𝑗, one 𝑗. And if we want to subtract the
vectors 𝐴𝐵 minus 𝐶𝐷, we’ve got two of the 𝑖s here take away negative two 𝑖s,
and we had negative three take away four of the 𝑗s. Then two take away negative two is
two add two, so that’s four of the 𝑖s. And negative three take away
another four is negative seven 𝑗s. So we’re adding negative seven
𝑗. So we probably not write plus
negative 𝑗, we just write four 𝑖 take away seven 𝑗.
So hopefully, you’ll be comfortable
using the 𝑖 and 𝑗 unit vectors just to represent the 𝑥- and the 𝑦-component of
any vectors that you come across now.
