 Okay, so welcome to the first lecture on the vectors course. This is the basics, vectors versus scalars, vector notation, addition and scaling, and properties. Alright, so begin at the beginning. Let's list some scalar quantities. Think about mass, duration, length, temperature, charge. These physical quantities are all well described with a single number. Really they just have a magnitude, although some of them may go negative, so it's a magnitude and a sign. But still, just a simple number is adequate to describe these things. How about vector quantities? What's what's different about vector quantities? Well, think about these things, force, velocity, and therefore acceleration or momentum. These things also have a strength or a magnitude. However, so let's put that down, they have a magnitude. However, they also have a direction. More than just a sign, they have a full-on direction in three-dimensional space. So it's not enough to know that a force is three newtons. I want to know in which direction is that force applied. And that then is the difference between a vector and a scalar quantity. We're going to think about how we manipulate them. Alright, so first off, the notation that we're going to use when we talk about our vectors. What I'm going to do is I'm going to use a symbol such as the letter A. So let's write that out. But I'm going to underline it. So an underlined symbol indicates a vector rather than a simple number. And when I need to specify that vector, I'm going to write it. So we're going to be three dimensional. I'm going to write the three numbers in a column form like this. Now if you haven't seen a vector specified before, what does it mean? Well, think of the Cartesian axes, the x, the x, y, z axes. Think in this case about coming out from the origin two in the direction of x and one in the direction of y and three in the direction of z. What we're going to do is we're going to think of our vector as an arrow. An arrow that comes from the origin to this point in space. And that arrow itself, whether or not it comes from the origin, that direction and that length of arrow is our visualization of the vector. So let me just change color to green and go ahead and draw the tip of my arrow there. There we are. So the vector is coming towards us out of the screen and it has those particular three components two, one, three. Other people may use other notations. For example, a line over the symbol A is commonly used. When people write out the components they may choose to do it as a row like this. Or even using pointy brackets like this. Now all these notations are basically getting at the same thing. You'll be able to read textbooks or look online and see these things and understand what they mean. But within this course of videos we're just going to use the notation that I've introduced above. So I'll erase those for now. Now the simplest thing that you might want to do if you have a couple of vectors is to add them up. So let's think about that vector addition. What does it mean? So let's give ourselves a second vector B. We'll make it 5 minus 2, 0 let's say. I want to add these two vectors together. So we'll write that out. I simply want to add A underline plus B underline. What does that mean? Let's just substitute in 2, 3. Add it on 2, 5 minus 2, 0. Now what we do is we simply add the first component of vector A to the first component of vector B and so on down the list. Very, very simple. So we're adding 2 plus 5. We're going to add 1 plus minus 2 and 3 plus 0. And we just tidy that up. So that's going to be 7 minus 1 and 3. Now how about scaling a vector? Okay, so what we can do is we can multiply a vector by a simple number and correspondingly we'll just end up multiplying each of its components. So let's take an example 3, 9, minus 12. What we notice is each of the three components is a multiple of 3. We can just take that common factor out in front and write this instead as 3 times 1, 3, minus 4. Same thing. Alright? Or equivalently someone might give us a vector that's already written in this form. It could be let's say 3 over 2 on to 2, 4, minus 4. Let's make it 1. Alright? And we can just multiply that in a component by component basis. So we just write ourselves a new column. Of course 3 times 2 is 3. 3 times minus 4 is minus 6. And 3 over 2 times 1 is 3 over 2. Okay? So there we are. We can scale our vectors by a number in this simple way. So with these definitions of addition and scaling, can we say anything about the properties? Okay, so if I have two regular numbers A and B, then of course A plus B is the same as B plus A. I'm not saying anything fancy here. It's as simple as, I don't know, 7 plus minus 3 is equal to minus 3 plus 7. Obviously it is. We know that. Now, if we think about the same statement for vectors A plus B, vector B, is it the same as vector B plus A? Well, it must be. Let's just write out an example, 7, 0, minus 1, 3, 1, 2. Is it equal to 3, 1, 2 vector plus the vector 7, 0, come on, 7, 0, minus 1? Of course it is because of the way we've defined vector addition as just being the addition of each element to the corresponding element. And this property is called being commutative. Okay, so vector addition is commutative. How about this second example? If we have three basic quantities, ordinary numbers, then if we have A plus B plus C, it's the same as A plus B plus C. It doesn't matter the order that we do them in. Is that going to be true for vectors? Well, of course, it is going to have to be true to vectors because the way we define vector addition is to add each component to the corresponding component. It's just addition. So this is also for vectors. Let's write down what we mean. We mean that vector A plus B plus C as a previously worked out thing is equal to vector A plus vector B and then add on C. It doesn't matter the order we do these things. All right. And there's a name for that property. It's called being associative, associative. All right. So vector addition has that property also. Now let's think about our scaling property. If we have ordinary numbers again, then we could take some scale factor K and multiply it into A plus B and it would just give us K times A plus K times B. Again, I'm not saying anything that isn't utterly obvious here. Say for example, I don't know 2 into 1 plus minus 3 is equal to 2 times 1 plus 2 times minus 3. Of course it is. So how about for vectors? Is it true that some scale factor K times the sum vector A plus B and let's stress that this scale factor is just a pure number? Then yes indeed, it's going to be just K times A plus K times B. So just to stress what we're doing here, let's copy down this sum of 2 vectors we were playing with up here. This 7, 0 minus 1 thing plus 3, 1, 2. Put it inside curly brackets maybe for a variety. It doesn't have to be curly brackets. Multiply it by some factor. Let's have 3 over 2. Had that before. Unimaginative. There we are. What's that going to be? It's just going to be 3 over 2 times the first vector 7, 0, 1 and then plus 3 over 2 times the second vector 3, 1, 2. Okay. So everything as you kind of would expect it works out. It must. And this latter property is called being distributive. So scaling is distributive over addition. And that's the end of our first video.