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2D Vector Basics - Overview

In mathematics, a vector (from the Latin vehere "to transport") is a geometric object that has a magnitude (or length) and a direction. Vectors can be added to other vectors according to vector algebra, and can be multiplied by a scalar (real number). A vector is what is called a "linear" quantity; its magnitude is independent of the chosen basis, and is determined by the component vectors' components. A vector can be pictured as an arrow (or a directed line segment) with a definite magnitude and direction, and having a definite position relative to a chosen origin. The magnitude of the vector is the length of the arrow. The direction of a vector is the direction of the arrow.

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Video Transcript

Okay, so this is gonna be a video about two D factor basics. And so, first of all, let's go ahead and talk about what a vector is. Vector is just something that has a direction and aptitudes. The two components, um, of the vector are gonna be magnitude and direction, so it's basically amount in a particular direction. So then with these two components, um, you know that a factor is going to be basically just represented by something that looks like an arrow because basically the length of the arrow would equal the magnitude and then where the direction is pointing, so the point would be the attraction. So then, in terms of drawing that factor in a two D plane, all we would have to do is John or plain and then typically start from the origin. So, for example, I might have a factor that looks something like this. Or maybe even something like this, Maybe something like this. So these all have different directions and different magnitudes because they're pointing in different ways, and they have different lengths which represent the magnitude. So then, from here, we know that within once we drafted like this it's pretty easy to see. So let's just take that first one, for example. So at that point, it's pretty easy to see that we could draw a right triangle like this, at which point this vector will have an X component and a white component. So if I wanted to take this vector, let's just say that this vector had a length of 13. Then the X had, um component five. And why are the component of 12? That's like saying that the vector is going five to the right and 12 up to form an overall vector of 13. So when we draw it out, what we would do is have five in that direction. And then the important thing here is to put the factors together, head to tail. So here's ahead of the arrow. And then here's the tale of Vector. I'm putting them right together so you don't want to ever put the factors like this or like this. You want to make sure that they're going head to tail. That's it, too, in whatever direction that maybe so then from here, once we've drawn that head to tail, that's when you would draw the resulting factor, which is this, and that would be my 13. So then, in terms of writing that factor out, I might say that I have the point. So let's zoom out of this. So then, from there, one way to write it out is to say that this is a factor. RS, this is Victor RS. It would have our component and an s component. So, for example, with vector rs, we would know that you would have a certain point and point for our Let's just say that we are had a point. Mhm negative to come eight and s had a coordinate of eight comma. Negative too, basically just flips. Then what you would do is you would first start out by applauding those points on the graph *** to come. Eight is something right here. Um and then eight common native to or something like that. Then you would connect the vectors. And in this situation, it's very important to note that rs um it starts from our and goes to s because the vector indicates that it's going from left to right in that direction. So our goes first and then s goes first, later meaning that the air goes this way. If it was s are we would have the air going in the opposite direction since it starts with s and ends up our So be very careful with the directions because they do matter. So then from here, Um, basically, what you would do with that is you could either find an X component by component, because all you have to do split it into a right triangle and then find this when the extractions measure, which is 10, because going from negative 2 to 8 is 10 and then from eight to negative two, you're going negative. 10. So you have 10 and negative 10. So then this resulting factors magnitude would just be the square. So just kind of like using Pythagorean theorem than 10 squared was 10 squared becomes 200 and so you would square root that. So this value right here, the magnitude would become the square root of 200. That's how you would find the magnitude of that. So then another thing that you can find other than the magnitude is the angle between the measures. If I wanted to find this data value, that's really easy. Because although it seems tricky because we're dealing with factors, it's really not because you could just think of it as a right triangle, at which point you're using Pythagorean Theorem or Pythagorean Theorem. And so gotta So then, according to Sakata, I could just use tangent. You can use anything. Really? Because we're not. This is time. This is 10. This is Route 200. So if I use tangent, I know that tangent is equal to opposite over adjacent on. Then, from there I could say 10/10. So tangent this one attention equals one. So I know data is going to equal 45 degrees. So in that way, um, once you can connect your vectors and break it up into the components of the X component and the white component, you could easily find your ex component your white component as well as the magnitude of the vector and also the angle between the vectors. Uh huh. So then, from here, let's talk about the formats of writing factors. Then one thing that you could do to represent a factor is to write it in the format of like, rs like we talked about? Yeah, Um and then you would be able to know, Put it in this parentheses format where you put the X component on the top. Then why keep going on the bottom in vectors? We can also represent that as I and J. And I am Jay. I know that sounds kind of familiar, and some people might not like that very much. But the eyes basically just a stand in for X and J is basically just a stand in for Why So then, for this you could cook it. So if the vector was 34 and you would know that the I is three and the J is four and so you would be able to plot it so three to the right and then four up would make this kind of factor. So you have this 345 triangle that looks like that again. If we were to draw all the arrows, that's where there would be because we have head to tail, head to tail. Then another way of writing this out is to put it in component form. Which component form is just represented by this triangular bracket type situation or you have the X component and then the y component. Same thing I could put a comma Jake component. It's another way of representing it. And then so that would be component from And then within this component form again. If we review this, the magnitude will be represented by thes things that look like absolute values. So I would say the magnitude, uh, Dr R S is equal to five. So that's another thing that you could do. So then just keep in mind the different forms that you could have, and then you could have the eyes and the jays instead of exim eyes, and so you could also represent it out as an equation. So for this factor, you could also rewrite it as three I plus four j because that's what's going on. It's like we have three in the extraction and therefore in the right direction. If we had something like negative three I plus four j, what would be happening is, instead of going to the right, we would be going to the left by three and then still up by four. So keep in mind the signs, and that if it's a negative for the eye that's gonna be going to the left. And if it's negative for the J, it's gonna be going down as opposed to up.