Sherpa

A vector has both size and direction and they can be added to, subtracted from and multiplied by a scalar. Most geometrical problems encountered in everyday life can be solved using vectors.

A vector describes a movement from one point of origin to another. A vector quantity has both direction and magnitude whereas a scalar quantity only has a magnitude which is a fancy word for size. Vectors are represented by lines with an arrow in the centre of them.

Okay. Guys. We're going to be discussing vectors in this video vectors are a very useful part of mathematics used in a wide range of engineering scientific settings for example. If you're calculating a force field like an electromagnetic force field each point in space would be given a vector which shows the strength and the direction of the force that you would experience at that particular point so what is a vector? Well, a vector is an entirely different kind of object to the usual objects that you're used to in maths the normal objects that you're used to are scalars like normal numbers and all they have is a magnitude, a size for example, five five is a number, it's a scalar, it has a magnitude a size and the size is five that's it.

All of the numbers that you know. All of the algebraic symbols for example. If you have x squared equals nine and we figure out that x is equal to plus or minus three the x's are also scalars because in the end they're just a number in this case. Plus or minus three a vector is an entirely different mathematical object that you've never seen before it's a different thing it has both a magnitude a size. Like a number but it also has a direction numbers don't have directions I suppose you could count the plus and the minus as a direction but vectors have a 360 direction they can be in any direction so if you take a look here in this coordinate grid we've got two points A and B and the arrow drawn between point A is at three four and point B is at five seven now the vector goes two along and three up and in order to demonstrate this direction of this vector we show the vector as a vector column and what that says the vector column just like with a normal coordinate grid where you have your coordinates x comes first horizontal and then vertical along the corridor up the stairs as you were taught in primary school the x is at the top.

The Y at the bottom so that's two to the right and three up if you had a vector column. For example. Negative two three. That would mean two to the left because it's negative two and then three up so it would be in this sort of direction so that's how a vector column demonstrates to you the direction the size of the vector which we're not going to look into right now the magnitude of the vector is the length of this line in this case this would be a two. This would be a three you'd have to use Pythagoras's theorem and then find that the length of this squared is two squared plus three squared and then square root it because this ends up as a right angle triangle here but we're not going to do that just going to tell you that the magnitude of a vector is its size.

We're going to be concentrating more on the directions of vectors and how they work.

So here we have two vectors A and B. Now there are a few different notations for vectors. This ad is this vector here with this arrow going to the right. It's 1234 to the right. So this is notation in mathematics notation.

It's ad, but we can also give it as a small letter underlined, which is how we write it on the paper in textbooks. They won't do it as an underlining, they will do it as a bold, like so bold letter A. So there's the main three different types of notation. You see A to D with an arrow and that'll be in capitals. You'll see a small letter A with an underline written down on paper and in textbooks where they're printed, you'll see a bold letter A instead of the underline.

It's all the same thing. So right now we have this vector. We're going to use the underline notation from now on. We've got this vector A and this vector B. A is zero up and down and four to the right.

So it's four to the right and zero up and down. So it's the vector 40 B is two to the right and three up. So it's the vector two three. Now if we want, for example, the vector A to C, what we do is we find a path from the beginning A to the end C, going through different letters. So where's our path?

Well, our path is here, it's A to B plus B to C. Just imagine yourself as a little stick figure guy walking along these lines and that's all it is. If I want to go from A to C, then I walk from A to B and then I walk from B to C. Now what is B to C as a vector? It's not written on there.

Well, this is the interesting thing about vectors. It doesn't matter where they are, it only matters what direction and size they are. That's the only thing that defines a vector. So because this is four to the right and zero up and down, this is also the vector A and this one two to the right and three up is also the vector B. So this is clearly a parallelogram, got two parallel sides given by the vectors B and two parallel sides given by the vectors A.

So what is the vector here? AC.

Well, the vector AC is the vector AB plus the vector BC. That's our path to go from A to C, we go from A to B and then B to C. We know in terms of these underlying vectors that that is AB is just the vector B and BC is the vector A. And so that's it. The vector A to C is the vector B plus A.

It doesn't really matter the order that you add things up. So that's A plus B because you can go to the right and up as well. It doesn't matter which path you choose, as long as you get there, it will be correct.

So adding and subtracting vectors has an interesting geometric interpretation. So take this triangle here, ABC. Now if I want to go from A to C, I'll go from A to B and B to C. That's one possible path I could take.

A to C is A to B plus B to C. Okay. So that's all vector addition is if I want to go from one point to another point, I find a third external point and then go from my initial point through the external point to my final point, and then I get my proper vector notation for that. So we have AC, is AB plus BC. Now if on the other hand, I wanted to go from B to C, still just going from an initial point B to a final point C, I'm going to have to go against the vector AB here, I'm going to have to go in the opposite direction.

And what happens there? Well, an opposite vector, we know that a B is two, four to go in the opposite direction, where the arrow would be backwards. Here you would go two to the right and four down. So that's BA is two to the left rather, and four down, negative two and negative four. So in other words, to go in the opposite direction to a vector, you simply make it negative.

That's all. So what do we have here to go from B to C? We go minus AB plus AC.

And we already figured out what AC was, it was AB plus BC. Let's write this out. There we have AC minus AB, and if we plug AC in, we'll find that this does work out. And so there we go. We have a notation here for subtraction of vectors and that's the geometric interpretation.

To go from A to C, we can add vectors in this direction, or if we're going against a vector, we can subtract. So final point to make here is if I want to go from A vector B to A vector C, I can take a third point A here and I can take the final position vector from A which is AC minus the initial position vector from A which is AB. And that will be my answer. So one last example, what is the vector from A to B?

If I take a third external point C, then it's the final vector which is CB subtract the initial vector which is CA. So to go from A to B, it's the final vector from the origin C from our start point CBCA. And if you plug it in, this will work.

Okay, let's do a proper vector problem here and we'll see how this all fits together, because so far this has just been some collection of thoughts on vectors. Before we move on to this, there's one point we need to emphasise. If I have a vector, let's say this one all the way along here, A, which is one, two, three across and 12345 six up. So A is 36. If I wanted to half that vector and do the vector that's exactly half of A, then all I've got to do is half the coordinates.

So it would be 1.53 which cheques out one and a half would be here, three up here. So that works. So in other words, if you want to change the size of a vector, all you have to do is multiply by that factor. Another thing that can be used to prove is when two vectors are parallel. So for example, that vector we just showed, there a half A, we can prove that it's parallel to the full vector A, that's a half A just to this point because a half A is equal to some constant, which in this case is a half times A.

And if it is only different by a constant, then that means all we've done is change the size, the magnitude of the vector. And so those two are parallel, they lie on the same line there. That's going to be useful in this question.

So here's the main problem. We have a parallelogram here. M is the midpoint between B and C and N divides the line OB into the ratio, two to one. Question one is to find O to n, o to M, eight n and eight m. Question two is prove that an and M lie on a straight line.

It's a very common vector question, this one. So let's start by finding onoman and a m. The first thing we do is we find the easiest vectors on here. So that's just A and C. So let's just start off by giving those A, as we've seen, is three six and C is 12345.

Okay? And this of course will be C here and this will be A up and down there. So to go from O to n we need to find a path. Well, we can first of all go from O to A, then from A to B, then from B to n. But we still don't know where B to N is.

So we're going to need to know this OB line first and then use the ratio where this has been split into a ratio of two to one. So let's do that. We've got A and C. Let's call this one O to B.

And we know that O to B is A plus C. O to N is going to be two thirds of it because it split into the ratio of two to one. Two parts. One part means this whole line is three parts. So O to N is two thirds of O to B.

So O to N is two thirds of O to B.

Label that on there. B. So we know that B is A plus C. So we've got two thirds A plus C.

If we're going to give our vectors in terms of A and C, then we're done because we found our vector two thirds along the way of this slide. Now to go O to M, we need to find a path again from O to M. So let's go. I would say the easiest one is from O to C and then from C to M, where M is the midpoint of this vector, C to B, which is also the vector A. So o to M is OC is C plus a half a and there we go, we're done next, A to N from going from here to there, finding a path, we'll go from A to B, which is C, and then we'll go one third of the negative of B.

And that will get us from here today. Because remember, B was in the direction of bottom left, top right. So A to N is equal to C along here, plus B to N, which is plus one third of the negative. I'll do that differently. I'll just say minus rather than plus.

Then it's minus a third rather than plus a third if it's going in this direction, minus a third of B, where B was A plus C.

There we go, A plus C. So simplifying this out. We have C subtract A third A subtract A third C. Which simplifies to two thirds C minus.

And the last one. A to M.

We need to go from A to B. Which is C half of negative A. So minus half A. And then we're done at M is C.

So now that we've got all three. All four of these sorted. We can answer question part two. Prove that an and M lie on a straight line. What we're looking for here is to show that the vector from A to N is some constant multiple of the vector A to M.

That will prove that they are parallel. And since both of those lines, those vectors A to N and A to M contain this single point A, then we have that they are on the same straight line. So let's show that we have A to N, which is our two third C minus a third A. And we want to show that it's some constant multiple of A to M, which is C minus half A. So can we think of how we might do that?

Well, A to M is bigger that's A-C-A to N is two thirds C. So let's try multiplying by two thirds. That looks like that's going to work out. Just going to assume this and we'll see if it works out. So let's say A to N is two thirds A to M.

So that must mean that it's two thirds of C minus a half A, which would mean it was two three C minus a third A, which is exactly what we expect for eight N. So that is true. Eight N is two thirds of eight M. We've just shown that. And therefore, since A to N is two thirds, that means that A to N is some constant multiple of A to M.

And since A is shared by both of them, they lie on the same straight line. This here, this line proves that an and A M are parallel. And the fact that they both share the point A means that they lie on a straight line.

Okay, time for you to have a go at your own vector question here, similar to the one we just did before. Just an extra step up in difficulty. If you think you've got the correct answer, please put your answers in the comments section. My name is Adam Tudor. For sherpa.

I can give the answer to this question, as can any math tutor for Sherpa. I'd love to see your comments on this. Let me know if you think you've got it right. Have a good day, ladies and gentlemen. Goodbye.