In this video, we’re gonna talk about what vectors are and the notation that we
use to represent them.
Now this varies a little bit from country to country. So we’re gonna show you very different ways
of showing the same thing, using different notation. And you’ll just have to pick the-the flavor
of notation that works where you are.
𝐴 vector is a set of numbers which can be represented in a suitable space by a line segment
with a specific length and direction. In this case, we’ve got a two-dimensional vector which
represents a translation of three units in the 𝑥-direction, so we’re going positive three along
the 𝑥-axis, and negative two units in the 𝑦-direction, so we’re going
[negative two] in the 𝑦-direction. So to go from 𝐴 to 𝐵, we’re doing three in the 𝑥-direction
and negative two in the 𝑦-direction.
Vectors were used by astronomers back in the seventeen hundreds as a way of recording and
tracking movements of planets in the night sky. The word itself comes from Latin, meaning carrier.
So try to think of vectors as meaning, what direction and how far should I move if I want to be
carried from 𝐴 to 𝐵 in a straight line. Now think that that set of instructions can be applied to
any start location to reenact the same relative movement. So in this case, we’ve just moved a
certain distance in the 𝑥-direction without really changing much in the 𝑦-direction. So that same
set of instructions could be applied from different start and finish positions.
When we draw vectors, we use a line segment with a specific length and an arrow, so that we can see
the implied direction of travel. In definitions, you will often see that vectors have an initial
point and a terminal point. But remember that you can pick up the vector and place it anywhere
in space to represent the same relative movement, so long as you ensure that it’s got the same
length and orientation. So in fact, the initial point and terminal point are not actually fixed.
They just help you to calculate the direction and the distance of the movement. So if I add the
coordinates of 𝐴 and 𝐵 to my diagram, I can see that I was moving four in the 𝑥-direction and
two in the 𝑦-direction. So the vector is the combination of those two components, the 𝑥-component
and the 𝑦-component.
Now we’ll see what all those numbers mean, and we’ll think about vectors a bit more deeply in just
a moment. But first of all, let’s ask the question why should we use vectors in the first place.
We can quantify lots of things with just a number and sometimes with a unit. For example, mass, area
volume, length, they can all be represented by a number and sometimes a unit. For example, ten
kilograms, twelve square centimetres, a thousand cubic centimetres, and so on. These are all called
scalar quantities. Now vector quantities have an extra aspect of them, direction. So they could
describe, for example, a translation of a certain magnitude in a certain direction. So the
translation from point 𝐴 to point 𝐵, for example. They could also represent, say, a velocity of
a certain speed in a certain direction. And likewise, they could be used to represent acceleration,
or even force, applied in a specific direction. So vector quantities are all about situations
where we need a number to represent the magnitude of something, but we also need to know
specifically about the direction that it’s operating in.
So what notation do we use for vectors? Well we’ve seen that graphically, we can use a
line segment of a particular length orientated in a particular direction. But we can also use
letters and numbers to represent vectors, although there are severn- several different
conventions, depending on where you live. So let’s think about two-dimensional space, to start
with, but this can easily be extended to 3D or even higher dimensions, just by adding extra
So one way is called component form. Here we write the amount of translation in the
𝑥-direction first, then the amount of translation in the 𝑦-direction second. There’s three popular
formats, unfortunately. One’s got angle brackets, one has this vertical column vector format, and
another looks just like coordinate annotation. If we want to give our vectors a name, then one
popular and useful notation is to label it using its initial and terminal points, with an
arrow above showing which is which. The letters are usually uppercase, and the initial point is
first and the terminating point is second. Depending on where you live, you’ll either use an arrow
or a harpoon style half arrow above. So let’s look at those now.
So for these two, this usually in tight written format, you would have capital letters 𝐴 and 𝐵
in italics, with this harpoon style arrow above them. And for the column vector notation, that’s
normally associated with nonitalic uppercase letters with a normal arrow showing the direction
from 𝐴 to 𝐵. Hopefully, you’ll recognize the system that’s used in your local area from your
textbooks or your school work. So if we wanted to represent the vector going from 𝐵 to 𝐴, we
could use any of these formats over here on the left. 𝐵𝐴 is negative three, negative four
butog- because to go from 𝐵 to 𝐴, we’ve had to go negative three in the 𝑥-direction and negative
four in the 𝑦-direction. So the order of the letters, the direction of the arrow, the sign on
the numbers, are all crucially important when it comes to writing our vectors.
Now just to add to the confusion, if we just wanted to assign a particular letter to a-a vector,
rather than specifying its start and end point, we can do that as well. But again, there are lots
of different conventions that we can use. Now this is actually quite difficult, when you’re writing
this down to represent this on the screen. But typically, you might use a bold version of that
letter, could be uppercase, could be lowercase. So if we were to call our vector 𝑣, it would be
bold 𝐯 or italic 𝑣 is equal to three, four in this particular notation.
Now people who use the column vector notation also use a bold letter to represent the name of
a vector. So if we wanted to call the vector 𝑣, it would be a bold letter in typeface. But when
we handwrite it, we might write it as a normal letter 𝑣 in handwriting but with a- an underline
underneath it, to show that it’s a vector. So that’s just a way of helping you to write vectors
in handwritten notation.
And the people who use the coordinate notation for vectors, again, would use a bold letter, but
they put one of these little harpoon arrows over the letter to represent the fact that it’s a
vector. Or, they might even use an italic version of that letter with a harpoon over it to
represent the vector, sometimes.
So it sounds a bit confusing when we’re talking about it like this. But you’ll know from the
context of the questions that you’re looking at, what notation they’re using and that they mean
vectors. So let’s take a look at a couple of examples of vectors.
So vector 𝐴𝐵 is, well we’re moving five in the 𝑥-direction to get from 𝐴 to 𝐵 and we’re moving
positive three in the 𝑦-direction, so our vector is gonna be specified as five [five, three].
So there are the three different ways of writing that, the journey from 𝐴 to 𝐵. In all cases,
we’re adding five to the 𝑥-coordinate that we had at point 𝐴 to get the coordinate the we
had at point 𝐵. And we’re adding three to the ac- to the 𝑦-coordinate that we had at
𝐴 to get the 𝑦-coordinate that we’ve got at 𝐵. So that’s the translation five, three.
So now let’s consider the vector in the opposite direction from 𝐵 to 𝐴. How would that look? This
time, we’re doing negative five in the 𝑥-direction and negative three in the 𝑦-direction. We’re
going to the left and downwards. So we’d write that as 𝐵𝐴 is negative five, negative three in
whichever of those formats that you choose. Now since each component of our 𝐵𝐴 vector is the
negative of the component that we had in the corresponding 𝐴𝐵 vector, we can say that 𝐵𝐴 is equal
to negative 𝐴𝐵, or 𝐵𝐴 is negative 𝐴𝐵 in this format. So what we can see is a negative of a vector;
we’re travelling exactly the same distance but wed- we’re doing it in exactly the opposite direction.
And remember, we also had the option of just labelling our vector using a single letter. So we
could’ve had a bold 𝑢, or an underlined 𝑢, or a 𝑢 in bold with a little harpoon arrow on top. So
instead of using this notation, we could’ve used our individual letters instead. All means the
same thing and they’re all, likely, things that you’re going to encounter in questions, when you’re
looking at vectors.
Now in this example, the translation from 𝐶 to 𝐷 involves adding six to the 𝑥-coordinate
but subtracting four from the 𝑦-coordinate, to get from the 𝐶’s 𝑦-coordinate to
the 𝐷’s 𝑦-coordinate. So here are three different notations: angle brackets, column vectors, and
what looks like coordinate notation — six, negative four — and the different ways of writing 𝐶𝐷
with a different arrows or harpoon arrows over them. Alternatively, we could’ve labelled 𝐶𝐷
with an individual letter of its own either a bold 𝑣, or a 𝑣 underlined, or a bold 𝑣 with a harpoon over
it. And we could’ve replaced our 𝐶𝐷’s with those letters. So a bold 𝑣 is available in all
cases. Sometimes, you can just write a nonbold 𝑣 with an underline, and sometimes you need the
bold 𝑣 with a little harpoon on top of it.
So now we can represent vectors in various ways. Let’s also think about multiples of vectors.
A vector 𝑤 represents a translation of three, positive three, in the 𝑥-direction and negative
two in the 𝑦-direction. Then two 𝑤 would simply be a translation twice as great in each direction.
So you can picture this as two 𝑤s placed end-to-end, one after the other. And all we’ve done,
because we got two 𝑤, we’re doubling the 𝑥-component, we’re doubling the 𝑦-components. That
becomes six, negative four in each of these formats. And three 𝑤 will just be three copies of
the vector placed end-to-end. So we’d end up with nine, negative six.
And just again, with negative 𝑤, we’re taking the negative of each component. So the negative
of three is negative three, the negative of negative two is positive two. So in its component
format, negative 𝑤 is negative three, two.
So just to summarize that then, vectors have length and direction. Vectors represent a journey
from 𝐴 to 𝐵. They’ve got an 𝑥-component and a 𝑦-component. We’ve seen several different formats
that you might use, depending on where you live, to represent those components. And we’ve got a
choice of notations as to whether we use the initial point and the terminating point to label
them, or whether we use just a letter to represent a vector with a specific direction. We’ve also
seen that the direction is important. So, for example, vector 𝐵𝐴 is the negative of 𝐴𝐵. It’s got
the same magnitude but it’s operating in exactly the opposite direction. And we’ve also seen that
when we double or treble or multiply a vector by a particular number, we simply multiply the
individual components of the vector by that number. So vector two 𝐴𝐵 is two times the 𝑥 and two
times the 𝑦-component.
So hopefully after that quick introduction to vectors, you’re now familiar with various
different notations, what vectors mean, and why we use them.