# Vectors

by Jared Rovny

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00:01 Now that we have a better idea of how things move in 1 dimension, we're going to extend the ideas that we've discussed already into 2 dimensions.

00:10 In discussing 2 dimensional motion, we'll start with an idea of vectors.

00:14 We'll talk about 2 dimensional vectors and then discuss something that is very, very common and you might see in problems in the future which is projectile motion and how to handle projectile motion in a problem.

00:25 First we'd like to distinguish between what we call a scalar and a vector.

00:31 A vector is a quantity that doesn't just have a magnitude but also has a direction.

00:37 And a simple example makes this more clear.

00:39 For example if I asked you at each point in this room what is the temperature? You could say its 5 degrees, 10 degrees but you wouldn't say its 5 degrees that way or that way.

00:50 It's just a number. It's just 5 degrees, 10 degrees, some number.

00:53 And so at each point in this room, as you can see with these red dots, you could sort of imagine a temperature at each location.

00:59 But no direction associated with that temperature.

01:02 When this is the case, when you're talking about just a number with no direction, we call this number a scalar. On the other hand if I asked you right now, the air moving around in this room, which direction is it moving? Or how strong is the wind? You could tell me that at each point the wind is moving 1 kilometer per hour or 2 kilometers per hour in this direction or in this direction, and so at each point in this room I can think not just about how much the wind is moving, but in which direction the wind is moving.

01:30 Often when you see a vector like this wind is, you'll see an arrow over the letter.

01:37 So this arrow over the letter w, would mean that we're talking about the wind as a vector, a quantity with both a magnitude and a direction.

01:44 Here's a simple example of how a vector might work in 2 dimensions.

01:49 What we're looking at here is an airplane from a bird's eye view, and let's supposed this airplane is travelling slightly northeast.

01:55 The speed of the airplane is 500 kilometers per hour, say.

02:00 And the speed is just a scalar. It's just a number telling you how fast the object is going.

02:06 On the other hand, if I consider both the number and the red arrow, the 500 kilometers per hour and pointing northeast, I would call that the velocity of the airplane, and the velocity is a vector because I take into account not just the number, 500 kilometers per hour, but also in what direction that motion is in.

02:25 We can describe this more quantitatively in terms of vectors by describing the motion of the plane in the east direction, in the north direction separately. So for example, if I wanted to take this wind vector that is 500 kilometers per hour northeast, I could think about just the amount of the motion that is going east directly which would be 400 kilometers per hour and I could also ask how far is it travelling north or how fast and that is 300 kilometers per hour.

02:53 The way to write a 2 dimensional vector in this way, where we're just simply what we call break up the vector into its components, and that's what this is. You just take a vector and you think about the different perpendicular directions and you split it up or break it up into the 2 components, which we'll discuss how to do in a moment.

03:10 You would write your vector as you see here.

03:12 Velocity, I could call it a vector v with an arrow over it and I would write the 2 directions, the 300 and the 400, the north and the east separately but right next to each other.

03:23 So really this is just a book keeping convention.

03:26 We just take 2 numbers and sort of stable them together, just put them together.

03:32 We're just putting 2 numbers next to each other, and so there's no reason to be intimidated when you see a vector.

03:36 If you're looking at a vector and you're a little bit confused as to how to interpret it in your head, this is how to think about it. It's just different numbers put next to each other telling you how much of your vector is in each different component as we've seen with this airplane here.

03:51 Now that we understand what a vector is, as representing different components of a quantity in different perpendicular directions, we can ask what happens if we add a vector.

04:00 So for example, in this case if we have the airplane travelling partly north and partly east, what if we introduced a wind blowing to the east at 100 kilometers per hour? It would be quite a strong wind, but just for this example, if it was blowing only to the east, how could we find the new vector, the new direction of velocity that this plane would go? Unfortunately, this is pretty much exactly what you would expect if you're going to add vectors.

04:25 What you do as you can see here is the blue box and the green box we have are 2 vectors.

04:29 One is the vector of the plane's original motion, the 300 and 400 kilometers per hour north and east.

04:36 We're going to add to that the wind vector which has zero motion in the north direction, and so we have a zero in that first entry, and then a 100 kilometers per hour in the second dimension.

04:47 To add these 2 vectors, you just add them component by component.

04:50 So we add the north with the north, and the east with the east.

04:53 So we have 300 plus zero, for the north direction of the airplane after this wind is blowing on it, meaning it has a 300 kilometers per hour motion north, and then it still has that 500 since 400 plus 100 motion now heading to the east because of the addition of the wind vector.

05:11 And this is how you add vectors together.

The lecture Vectors by Jared Rovny is from the course Translational Motion.

### Included Quiz Questions

1. A magnitude and a direction
2. A location and a magnitude
3. A location and a direction
4. A position and a time
5. An acceleration and a time
1. A vector has both magnitude and direction, while a scalar is just a magnitude.
2. A vector has a specified location, while a scalar is just a number.
3. A vector is a sum of at least two scalars.
4. A vector only has direction while a scalar has magnitude.
5. A vector is used to represent positions while a scalar represents magnitudes.
1. They are added independently together.
2. They are added together to produce a single scalar.
3. They cannot be added directly.
4. They are perpendicular to each other.
5. They are combined before addition.

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