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Unit vectors and engineering notation

Using unit vectors to represent the components of a vector. Created by Sal Khan.

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Video transcript

What I want to do in this video is show you a way to represent a vector by its components. And this is sometimes called engineering notation for vectors. But it's super useful, because it allows us to keep track of the components of the vector and it makes it a little bit tangible when we talk about the individual components. So let's break down this vector right over here. I'm just assuming it's a velocity vector. Vector v. Its magnitude is 10 meters per second. And it's pointed in a direction 30 degrees above the horizontal. So we've broken down these vectors in the past before. The vertical component right over here. Its magnitude would be-- so the magnitude of the vertical component, right over here, is going to be 10 sine of 30 degrees. It's going to be 10 meters per second times the sine of 30 degrees. This comes from the basic trigonometry from sohcahtoa. And I covered that in more detail in previous videos. Sine of 30 degrees is 1/2. So this is going to be 5, or 5 meters per second. 10 times 1/2 is 5 meters per second. So that's the magnitude of its vertical component. And in the last few videos, I kind of, in a less tangible way of specifying the vertical vector, I often use this notation, which isn't as tangible as I like. And that's why I'm going to make it a little bit better in this video. I said that that vector itself is 5 meters per second. But what I told you is that the direction is implicitly given because this is a vertical vector. And I told you in previous videos that if it's positive, it means up and if it's negative, it means down. So I kind of have to give you this context here so that you can appreciate that this is a vector that just the sine of it is giving you its direction. But I have to keep telling you this is a vertical vector. So it wasn't that tangible. And so we had the same issue when we talked about the horizontal vectors. So this horizontal vector right over here, the magnitude of this horizontal vector is going to be 10 cosine of 30 degrees. And once again, it comes straight out of basic trigonometry. 10 cosine of 30 degrees. And so cosine of 30 degrees is the square root of 3 over 2. Multiply it by 10, you get 5 square roots of 3 meters per second. And once again, in previous videos, I used this notation sometimes, where I was actually saying that the vector is 5 square roots of 3 meters per second. But in order to ensure that this wasn't just a magnitude, I kept having to tell you in the horizontal direction. If it's positive it's going to the right, and if it's negative, it's going to the left. What I want to do in this video, is give us a convention so that I don't have to keep doing this for the direction. And it makes it all a little bit more tangible. And so what we do is we introduced the idea of unit vectors. So by definition, we'll introduce the vector i i. Sometimes it's called i hat. And I'll draw it like here. I'll make it a little bit smaller. So the vector i hat. So that right there is a picture of the vector i hat. And we've put a little hat on top of the i to show that it is a unit vector. And what a unit vector is-- so i hat goes in the positive x direction. That's just how it's defined. And the unit vector tells us that its magnitude is 1. So the magnitude of the vector i hat is equal to 1. And its direction is in the positive x direction. So if we really wanted to specify this kind of x component vector in a better way, we really should call it 5 square roots of 3 times this unit vector. Because this green vector over here is going to be 5 square roots of 3 times this vector right over here, because this vector just has length 1. So it's 5 square roots of 3 times the unit vector. And what I like about this is that now I don't have to tell you, remember, this is a horizontal vector. Positive is to the right, negative is to the left. It's implicit here. Because clearly if this is a positive value, it's going to be a positive multiple of i. It's going to go to the right. If it's a negative value, it flips around the vector and then it goes to the left. So this is actually a better way of specifying the x component vector. Or if I broke it down, this vector v into its x component, this is a better way of specifying that vector. Same thing for the y direction. We can define a unit vector. And let me pick a color that I have not used yet. Let me find this pink I haven't used. We can find a unit vector that goes straight up in the y direction called unit vector j. And once again, the magnitude of unit vector j is equal to 1. This little hat on top of it tells us-- or sometimes it's called a caret character-- that tells us that it is a vector, but it is a unit vector. It has a magnitude of 1. And by definition, the vector j goes and has a magnitude of 1 in the positive y direction. So the y component of this vector, instead of saying it's 5 meters per second in the upwards direction or instead of saying that it's implicitly upwards because it's a vertical vector or it's a vertical component and it's positive, we can now be a little bit more specific about it. We can say that it is equal to 5 times j. Because you see, this magenta vector, it's going the exact same direction as j, it's just 5 times longer. I don't know if it's exactly 5 times. I'm just trying to estimate it right now. It's just 5 times longer. Now what's really cool about this is besides just being able to express the components as now multiples of explicit vectors, instead of just being able to do that-- which we did do, we're representing the components as explicit vectors-- we also know that the vector v is the sum of its components. If you start with this green vector right here and you add this vertical component right over here, you have head to tails. You get the blue vector. And so we can actually use the components to represent the vector itself. We don't always have to draw it like this. So we can write that vector v is equal to-- let me write it this way-- it's equal to its x component vector plus the y component vector. And we can write that, the x component vector is 5 square roots of 3 times i. And then it's going to be plus the y component, the vertical component, which is 5j, which is 5 times j. And so what's really neat here is now you could specify any vector in two dimensions by some combination of i's and j's or some scaled up combinations of i's and j's. And if you want to go into three dimensions, and you often will, especially as the physics class moves on through the year, you can introduce a vector in the positive z direction, depending on how you want to do it. Although z is normally up and down. But whatever the next dimension is, you can define a vector k that goes into that third dimension. Here I'll do it in a kind of unconventional way. I'll make k go in that direction. Although the standard convention when you do it in three dimensions is that k is the up and down dimension. But this by itself is already pretty neat because we can now represent any vector through its components and it's also going to make the math much easier.