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Vector Basics Overview

In mathematics, a vector (from the Latin word "vehere" which means "to carry") is a geometric object that has a magnitude (or length) and direction. A vector can be thought of as an arrow in Euclidean space, drawn from the origin of the space to a point, and denoted by a letter. The magnitude of the vector is the distance from the origin to the point, and the direction is the angle between the direction of the vector and the axis, measured counterclockwise.

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Video Transcript

Okay, So a vector is a quantity that has two things. It has a magnitude and a direction. And so maybe the easiest way to see what a vector is is Just think about some examples, So some examples of vector quantities are forces. So if I apply a force to an object, of course it has a magnitude. I can have a greater force or a lesser force, but then it also has a direction. I can apply the same magnitude of force in different directions, and I'm gonna get different results. Another example would be philosophy. So if I go get in my car or my bike, whatever I can travel with a certain speed, that would be the magnitude of the velocity. But then I also have a direction in my direction. May change eso my velocity is gonna change. And so that makes it a victor quantity because it has a magnitude in a direction. So what about quantities that aren't vectors? All those would be scale er's And so some examples of scaler quantities would be temperature. So if I ask what's the temperature outside? I'm just going to get a quantity 28 degrees and there's no direction in that temperature is just That's the temperature. It's just a quantity. So another example of a scaler would be energy. So if I have an object, a ball or something like that, and it's gonna have a certain amount of energy, whether that's in the form of kinetic or potential, but the quantity of energy it has has nothing to do with any direction. So that would be a square, a scaler quantity. Okay, so if we're working in some sort of coordinate system, so if we have, say, the X Y plane or this could be, um, three dimensional space so we could also have a Z axis if you want to, then we can represent a vector quantity as in era. And now what we're gonna use to describe vectors are lower case letters like V with a little arrow on top. So this vector V Is this represented in this coordinate system as an arrow and it has components, so I can write the component vector and component form. And so thio each coordinate access. I have a component. So we has an X component, which I put Visa. Becks has why component and it has Z component like that. And of course, if I was just working in the X y plane, then maybe my vector only has two components of x component and a like a boat. But really, everything we're gonna talk about is the same whether you're thinking about having three components or just two components. All right, so let's say I have two vectors written in component for him. And just for now, I'm just gonna consider them in space of having 33 coordinates or components. Everything we're going to talk about works the same if we just have to components. So now if I have another victory, you has its three components. What I can do is I can add two vectors together, and this is really one of the most useful things about writing vectors and component form is because it gives me an easy way toe. Add them together. Soto add two vectors. What do I do? I simply add the components So the new X component is going to be the X component of fee plus the x component of you the new Y component. Just gonna add the why components and Then there's the component. Just add the Z components and we can actually visualize this in terms of errors. So if I have my Dr V right here and I have my vector you so I'm going to draw them, starting at the same point that so maybe I can think about this is some object on applying to force this way and force this way? Well, then where is the result? Enforce? Gonna be so what I do is I just take a copy of you. So let me put my copy of you up here. So this is awesome you. Now, the cool thing about vectors is that they really don't have to be based at any point. I can translate them and they're still the same vector. And then I can take my vector V and translated over here and put it at the end of you like that. This is still the specter of EU have just moved it over. And then the resulting vector is gonna be the diagonal. This parallelogram that's formed right here. So this is going to be the vector you plus B in here. So in addition to adding vectors, I can also re scale vectors. And what I mean by re scaling is actually re scaling a vector by a scaler values of making it either longer or shorter. So I could do that. So if I have a scaler, see, so just a number two or three or whatever and I multiplied by a vector and what happens is that each of the components of V gets re scaled by the same scaler quantity. See. So if I double the vector, if I multiplied by two, then all of the components just get multiplied by two. So these were sort of the arithmetic operations with vectors I can add vectors on. I could re scale vectors. So let's talk a little bit about the magnitude of a vector. So we said that the vector was quantity that had a magnitude and direction. So let's first talk about the magnitude of a factor. V Well, this is also called so it's also called the length of the vector, So the magnitude of a vector V is also called the length, and it's denoted by thes bars around the Vector V. It's sort of like absolute value of V, but it's the magnitude of the length of the And what is that equal to? What's the square root of the some all of the components square. And this is sort of coming from like the distance formula. So if I have a vector, the here and I have all of it components of as, say, an X component, why component Z component like that, then the length of the vector. And here I just mean really the literal length of this segment here that's making up the vector is just given by the distance. From this point to this point in this coordinate system, the distance formula is exactly this. So that's the length of the vector. And this also gives me a way to talk about the direction of a vector V. And so what do I mean by direction? Well, I sort of mean the way that it's pointing, but to be more specific, the direction of the is going to be a unit vector And what I mean my unit vector. I just mean that it's a vector of length, equal to one well, pointing in the same direction. So how do I find this direction? Vector of the well, I'm gonna denote it by V, and I'm gonna put a little hat over V so v hat. It's just gonna be V. So that gives me the direction. And to make it length one, I just divide by the length. So the direction of the is a specter where I take b and I divided by the length of the And so the nice thing about this notation is that any vector V can always just be written as its length your magnitude times its direction, Yeah.