Vectors in 2 dimensions - what are they?
A free video tutorial from Mark Misin
Aerospace & Robotics Engineer
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Calculus + Engineering + PID: Functions, Limits, Derivatives, Vectors, differential equations, integrals: BEST CALCULUS
34:53:37 of on-demand video • Updated October 2020
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English [Auto] Welcome back. Today we're going to start learning about vectors and the first thing that I want to do. I want to explain to you the difference between scalars and vectors. So what is the difference between scalars and vectors. Well look you have some quantities in the world that you can measure for which you only need one number line. For example let's take mass right. Let's just take mass. So in order to describe mass you only need one number line. So we measure the kilograms for example and then we have zero kilograms. We can have five 10 and Sidor kilograms. Right. Or another thing would be for example money we could have. Dollars or euros or pounds. And then again we could have let's say minus 200 miners 100 0 100 200. So here here you would have something like profit and here you would have something like Los or how much money you have and your debt whatever your context is. And another scalar quantity would be distance right because what is distance distance is land covered. So we we can measure that in meters and what it really is it's how much land have you covered. OK so that's this distance. It doesn't matter where you are where you're going. Just how much land you have physically covered and you might have coverage 0 meters. You might have covered five meters 10 meters. Also you could have speed and you can measure speed in meters per second or kilometers per hour. It's up to you. And speed is how fast you cover the land. So if you're addressed at zero meters per second you can convert the land at 10 kilometers per second 20 kilometers per second. It's up to you now. All those quantities we were able to measure that using only a one dimensional number line right is just one dimension you should look at it. It's just one line. You don't need any more dimensions. However we also have quantities for which we need to use more than one number line to describe it. And in order to do that we need tools to capture all the information that happens on one line and on and on another number line. For example let's consider this to the plane. And on this to the plane we measure position we measure position of this human here. And also we have this little model airplane and we didn't know the position of the human P1 the airplane P2. Now notice that this is not a function here. A function is a mathematical relationship between two variables. However here we just measure position of objects. So this is the real world you have a house you have a you have someone here you have a model airplane. And we just use our we are using a rectangular grid system to measure the position of objects. All right. So this is not a function here. So in order to measure let's say the position of this guy here or girl we would say something like this. P1 OK we know that this person is 10 meters and we measured in meters. We know that it's 10 meters in x direction. So that's one line. However it's a two deep plane. So how do you know where exactly that person is that person might be here or here. Or hear Underground's you don't know. So you also have to specify. And the second dimension and for that we use vectors. The mathematicians have invented vectors and vectors are tools that enable us to capture information that require more than one number line. Right. So we have more than one number line and we use vectors to capture information. So the way we would describe this position here Position 1 we don't know vectors with this upper small arrow on top of the letter and then we say that ok p one is X and Y. So the extension goes first and then why then mention comes later and then we just say 10 and 0 meters. And now we have fully described the position of the person and the same thing with the airplane. I can just say to X Y and it would be minus 20 and 15 minus 20 in the extension and 15 in the wide dimension meters. So this is called a two dimensional vector. So these quantities here are right for which you only need one number line. You call them scalar and here this is a vector and essentially if you think about it then scalar is just a one dimensional vector. And of course you also need in a real life you also need to measure that position in three dimensions. So again this is not a function here it's just you're using a cubic grade system. You see there's a little cube here. So you have divided your world into little cubes and now you are measuring the position using the cubic grid system. So let's say that there's something here right. Maybe another airplane for example in this position. And as you can see you need three number lines to determine the position. So let's call it P3 to determine the position 3. You need an x number line you need y number line and you need the z number line. So for the X it would be 20 meters for the white would be 30 meters and for the Z it would be 40 meters. And the way you would write it you would write it very simply p 3 arrow x y z and it would be then 20 30 and 40 right 20 30 and 40. So this is a three dimensional vector that represents the position of an object in a three dimensional space. But we also have situations for which you need a lot more number lines not only three number lines. So you can have vectors that have a lot more dimensions than three dimensions. And of course you can't really represented graphically especially if you try to represent them geometrically. However you can have quite this foot for which you have many number lines. And in principle you can have as many dimensions as you want or need. Up until infinity. So that's really really depends on your context. And I'm going to give you an example of that.