Vector intro for linear algebra

a vector is something that has both magnitude and direction magnitude and direction so let's think of an example of what wouldn't and what would be a vector so if someone tells you that something is moving at five miles per hour this information by itself is not a vector quantity it's only it's only specifying a magnitude it's not we don't know what direction this thing is moving five miles per hour in so this right over here which is often referred to as a speed this is a speed is not a vector quantity just by itself this is considered to be a scalar quantity if we want it to be a vector we would also have to specify the direction so for example someone might say it's moving five miles per hour east let's say it's moving five miles per hour due east so now this combined five miles per hour are due east this is a vector quantity and now we wouldn't call it speed anymore we would call it velocity so velocity is a vector we're specifying the magnitude five miles per hour and the direction east but how would can we actually visualize this so let's say we're operating in two dimensions and what's neat about linear algebra is obviously a lot of what applies in two dimensions will extend to three and then even four or five six as many dimensions as we want our brains have trouble visualizing beyond three but what's neat is we can mathematically deal with beyond three using linear algebra and we'll see that in future videos but let's just go back to our straight traditional two-dimensional vector right over here so one way we could represent it as an arrow that is five units long we'll assume that each of our units are our unit here is miles per hour and that's pointed to the right where we'll say the right is east so for example I could start an arrow right over here and I could make its length five the length of the arrow specifies the magnitude so one two three four five and then the direction that the arrow is pointed in specifies its direction so this right over here could could represent visually this vector if we say that the horizontal axis is say East or the positive horizontal Direction is moving in the east this would be West that would be north and then that would be south now it's interesting about vectors is that we only care about the magnitude in the direction we don't necessarily not care where we start where we place it when we think about it visually like this so for example this would be the exact same vector would be equivalent vector to this this vector has the same length so it has the same magnitude has a length of five and its direction is also due east so these two vectors are equivalent now one thing that you might say is well that's fair enough but how do we represent it with a little bit more mathematical notation so we don't have to draw it every time and we can start performing operations on it well the typical way one if you want a variable to represent a vector is usually a lowercase letter if you have if you're publishing a book you can bold it but when you're doing it in your notebook you would typically put a little arrow on top of it and you usually there's several ways that you could do it you could lure say hey five miles per hour east but that doesn't feel like you can really operate on that easily the typical ways to specify if you're in two dimensions to specify two numbers that tell you how much is this vector moving in each of these dimensions so for example this one only moves in the horizontal dimension and so we'll put our horizontal dimension first so you might call this vector five comma 0 it's moving five positive five in the horizontal direction and it's not moving at all in the vertical direction and the notation might change you might also see notation and actually in the linear algebra context it's more typical to write it as a column vector like this five zero this once again the first coordinate represents how much we're moving in the horizontal direction and the second coordinate represents how much are we moving how much are we moving in the vertical direction now this one isn't that interesting you could have other vectors you could have a vector that looks like this it's moving let's say it's moving three in the horizontal direction and positive 4 so 1 2 3 4 in the vertical direction so let me see it might look something like it might look something like this so this could be another vector right over here maybe we call this vector vector a and once again I want to specify that it is a vector and you see here that if you were to break it down in its Mundi in the horizontal direction it is moving it is moving 3 or it's it's it's it's shifting 3 in the horizontal direction and is shifting 4 positive 4 in the vertical direction so and we get that by literally thinking about how much how much we're moving up and how much we're moving to the right when we start at the at the end of the arrow and go to the front of it so this vector might be specified as 3 3 4 3 4 and you could use a Pythagorean theorem to figure out the actual length of this vector and you'll see because this is a 3 4 5 triangle that this actually has a magnitude of 5 and as we study more and more linear algebra we can we're going to start extending these to multiple dimensions obviously we can visualize up to three dimensions and four dimensions it becomes more abstract and that's why this type of a notation is useful because it's very hard to draw for 5 or 20 dimensional arrow like this