 So one of the important ideas in mathematics is known as a vector and so let's say we start with some set S and it really doesn't matter what it is some possibilities. We might start with the set of real numbers When there's an infinite number of elements in this set We might take a set of the set of colors that a computer can display And there's a few million elements in this set or we might take the set of trustworthy politicians Now this in order for us to actually have anything to talk about we need to make sure that this set that we're talking about is Actually non-empty. So this may not actually be a viable set So a vector is an ordered list also known as a tuple of the elements of S So for example, we might take a four tuple of colors That's four colors in a very specific order. And so maybe that order will be our red G green blue and then red We might take a five tuple of integers That's five integers and we're going to choose those integers in some particular color in some particular order So minus one two five negative one and four and then we may take a three tuple of trustworthy politicians Well, I can't think of any but presumably they exist So a little bit about the anatomy of a vector We often designate vectors using either a bold face time or we draw an arrow over the name of the vector So we might talk about the vector V in caps or the vector a with an arrow over it Typically in print you see the bold face notation, but because it's hard to write bold face on a board We often see we often write it when presenting it with the arrow over top The components of the vector are usually going to be indicated by adding a subscript to Indicate their places within the vector So we might talk about the vector V And that's going to be the components V1 V2 V3 and so on up to the VN Where V subscript tells you that this is the first component second component third component and so on now from us To this course we're going to be focusing on vectors whose components are drawn from the real numbers And so if our components are n tuples that sets of ordered sets of n real numbers We say that the vector is in our and that's the real numbers That's the Cartesian product of n factors of the real numbers And so for example, we might take the vector 1 2 minus pi and that is a C1 2 that is a 3 2 pole in other words. This is a vector in our 3 So here's a useful idea, which is the interconnection between geometry and algebra It is very very very very very very It is extremely helpful to be able to navigate between algebraic and a geometric Interpretation of a mathematical object So for vectors in our end well on the one hand we could look at them algebraically a vector is an end to Components each of which belongs to the set of real numbers So this is an ordered set v1 through vn and Geometrically Well, how might we look at this? Well, the thing to notice is that when we describe a vector it looks an awful lot like how we describe a point So this looks like it's a point in our end However, if we want to make the most use out of this intercount activity, we might notice that we already have points So what we'd like is the vector should be something that is different from a point because we already have points So what can we do? Well, let's think about this the coordinates of a point tell us two things of importance They tell us where the point is But they also give us directions for getting to the point from the origin And so what this suggests is that we can get value added by introducing vectors by making Direction the important thing about a vector and We can draw these directions. We can indicate them as arrows So I'm going from here to there. And so here's my vector u and likewise We're here there here there and so on That's something that's useful if we're going to interpret vectors as directions Then well this vector u here and this vector r here They look very similar there seem to be giving the same directions go that way go that way And so we might say that you and r are going to be the same vector The spatial position of the vector is not important where we happen to draw the vector doesn't matter At the same time if we viewed vectors at directions He Q have to be regarded as different. He goes this way Q goes that way Even though in many ways these two look very similar they're going in different directions So we have to regard them as different vectors and Well, here's an interesting case here this vector r and this vector s do appear to be going in the same Direction, but they're drawn a little bit differently. And so there's a similarity between those two vectors But they're not the same vector and we'll talk a little bit more about why these are Different and how we can distinguish between two of them So more generally if I have two points be a Q I can designate the vector PQ with an arrow over it as the vector giving the directions from this point P to this point Q So for example, let's take the vector going from the point P to the point Q Let's find both PQ and QP So let's think about that vectors are directions So if I go from P to Q what has to happen? Well, let's look at our coordinates. So our coordinates of P one three negative one our coordinates of Q zero negative three two And what do I have to do? Well, if I'm going from P to Q The first coordinate has to go from one to zero. It has to be decreased by one unit My second coordinate has to go from three To negative three, which means I have to decrease the second coordinate by six units And then that third coordinate goes from negative one to two It's got an increase that third coordinate by three units And what this suggests is that this vector from P to Q has to be negative one That's my decrease of one unit plus six negative six That's my decrease by six units and then plus three. That's my increase by three units Now I can do the same analysis to find the vector QP But I could also just simply reverse the directions is this is how I go from P to Q If I reverse what I do then I can go from Q back to P So instead of decreasing my first coordinate by one unit I want to increase by one instead of decreasing by six I want to increase by six Increasing by three I want to decrease by three and that's a guess that my vector should be increase of one increase of six, decrease of three, one, six, negative three And we'll close with one final idea What happens if I want to go from Q to Q? In other words from a point to itself Well, we would change none of the coordinates So this would be a vector consisting of all zeros and this is actually a useful idea for later on This gives us what we call the zero vector and again That's going to be no surprise zero with an arrow over it or a bold face zero And this is a factor whose components are all zero