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 three [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 numbers.
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.