Welcome back, everyone. So in the last video, we got introduced to a vector, and we talked about how vectors are represented as these arrows in space. Now what we're going to be talking about in this video is ways that we can actually add vectors together. It turns out in future videos, we are going to learn that there's a way you can apply numbers to these vectors and add those numbers together. But for now, we're mostly just going to be focusing on ways that we can visually add vectors together. So without further ado, let's get right into things.

Now let's say we have vectors u and v, and we wish to add them together. It turns out there is a straightforward way of doing this using something called the tip to tail method. And this method is exactly as it sounds. You take the tip of your first vector, and you connect it to the tail of your second vector. So if I take these vectors and I move vector 'v' so it's connected tip to tail with vector 'u', notice that we've just connected these two vectors together, and all I did was take this vector v and shift it so it's up here. Now once you've connected these vectors tip to tail, you want to draw a resultant vector, and the resultant vector is going to go from the initial point of your first vector to the terminal point of your second vector. So if you connect the vectors in this fashion, this is going to give you the vector, u+v. So as you can see, it's actually very quick. All you do is connect the vectors tip to tail, draw this resultant vector, and that's how you can solve these types of problems.

Now let's say we instead wanted to subtract two vectors. Is there a way that we could go about doing this? Well, you might think this is significantly more complicated than addition, but it turns out it's actually not. Because recall that subtracting vectors is really just the same thing as adding a negative vector. And we've already talked about what negative vectors are in the previous video. So if I wanted to take v and subtract it from vector u, I could just make vector v negative. So u minus v would be the same thing as u plus negative v. And recall that a negative vector is simply going to be a vector that points in the opposite direction. So here we have vectors u and v, and if I wanted to connect these tip to tail, I'd connect this vector with u. But notice how I'm actually going to draw the vector so it goes in that direction, it has the same magnitude but opposite direction because we have a negative vector v. So we have u and negative v. I can connect these with the initial point of the first vector to the terminal point of the second vector, and this right here would give me the vector I'm looking for, u minus v. So as you can see, it's the same idea as adding vectors together. Just when subtracting vectors, you have to make one of the vectors negative.

Now to really make sure we're understanding this concept, I want to take a look at an example. And something else that's important about this example is it's actually going to tell us something interesting about the order that we do these operations for addition and subtraction.

So let's start with example a. Example a asks us to find vector v plus vector u. Now what I'm going to start by doing is drawing vector v. I can see vector v looks something like that. Now what I need to do is I need to use the tip to tail method to find v+u. I can see that we have vector u right about here. So what I'm going to do is connect these vectors tip to tail, and I can see the vector u is 7 units across and 1 unit up. So going here, I'll connect it to the tail of vector v. I'm going to go 7 units across and 1 unit up, and this right here is going to be vector u. Now at this point, what I need to do is connect these two vectors from the initial point of the first vector to the terminal point of the second vector. And this right here is going to be the vector v+u. Now I want you to notice something about the vector that we got. Notice that the result is actually the exact same thing that we got up here. We got vectors with the exact same magnitude. They're both the same total length, and they point in the same direction. So in this case, when adding vectors, whether it's v plus u or u plus v, it turns out the order actually doesn't matter for these vectors.

But does this also hold true for subtraction? Well, let's go ahead and try it. So let's move on to example b, where we need to find vector v minus vector u. Now I'm going to start by drawing vector v, and I can see that vector v is going to look something like this. Now what I need to do is I need to connect this tip to tail with vector u, but notice vector u is going to be negative. Vector u is 7 units to the right and 1 unit up. So what I'm going to do is reverse this direction. So we're going to go 7 units to the left, and then we're going to go 1 unit down. So this would be vector negative u. Notice how vector negative u has the same length or magnitude but it points in the opposite direction. Now in order to find vector v minus u, what we need to do is take the initial point of the first vector and connect it to the terminal point of the second vector. So drawing this vector is going to give me the vector v minus u. And notice something about this vector. The vector that we got when we subtracted, v minus u, actually is different than the vector when we did u minus v. Because for vector u minus v, notice that this points down into the right, and the vector v minus u points up into the left. It's pointing in an entirely different direction. So it turns out the order actually does matter when we're dealing with subtraction, but the order doesn't matter when we're dealing with addition. And I think this actually makes logical sense when you just think about the numbers. Because, for example, if you were to have 5 plus 3, this would equal 8. And 3+5 also equals 8. The order doesn't matter when it comes to addition. But if you have 5 minus 3, that equals 2. And if you have 3 minus 5, well, that equals negative 2. Notice that it's not the same result when we subtract, but it is the same result when we add. So the same rule that applies to numbers also applies to vectors. So that is how you could add or subtract vectors, and how the order does or doesn't matter depending on what operation you're doing. So I hope you found this video helpful. Thanks for watching.