Adding vectors algebraically & graphically

So I have two 2-dimensional vectors right over here, vector a and vector b. And what I want to think about is how can we define or what would be a reasonable way to define the sum of vector a plus vector b? Well, one thing that might jump at your mind is, look, well, each of these are two dimensional. They both have two components. Why don't we just add the corresponding components? So for the sum, why don't we make the first component of the sum just a sum of the first two components of these two vectors. So why don't we just make it 6 plus negative 4? Well, 6 plus negative 4 is equal to 2. And why don't we just make the second component the sum of the two second components? So negative 2 plus 4 is also equal to 2. So we start with two 2-dimensional vectors. You add them together, you get another two 2-dimensional vectors. If you think about it in terms of real coordinate spaces, both of these are members of R2-- I'll write this down here just so we get used to the notation. So vector a and vector b are both members of R2, which is just another way of saying that these are both two tuples. They are both two-dimensional vectors right over here. Now, this might make sense just looking at how we represented it, but how does this actually make visual or conceptual sense? And to do that, let's actually plot these vectors. Let's try to represent these vectors in some way. Let's try to visualize them. So vector a, we could visualize, this tells us how far this vector moves in each of these directions-- horizontal direction and vertical direction. So if we put the, I guess you could say the tail of the vector at the origin-- remember, we don't have to put the tail at the origin, but that might make it a little bit easier for us to draw it. We'll go 6 in the horizontal direction. 1, 2, 3, 4, 5, 6. And then negative 2 in the vertical. So negative 2. So vector a could look like this. Vector a looks like that. And once again, the important thing is the magnitude and the direction. The magnitude is represented by the length of this vector. And the direction is the direction that it is pointed in. And also just to emphasize, I could have drawn vector a like that or I could have put it over here. These are all equivalent vectors. These are all equal to vector a. All I really care about is the magnitude and the direction. So with that in mind, let's also draw vector b. Vector b in the horizontal direction goes negative 4-- 1, 2, 3, 4, and in the vertical direction goes 4-- 1, 2, 3, 4. So its tail if we start at the origin, if its tail is at the origin, its head would be at negative 4, 4. So let me draw that just like that. So that right over here is vector b. And once again, vector b we could draw it like that or we could draw it-- let me copy and let me paste it-- so this would also be another way to draw vector b. Once again, what I really care about is its magnitude and its direction. All of these green vectors have the same magnitude. They all have the same length and they all have the same direction. So how does the way that I drew vector a and b gel with what its sum is? So let me draw its sum like this. Let me draw its sum in this blue color. So the sum based on this definition we just used, the vector addition would be 2, 2. So 2, 2. So it would look something like this. So how does this make sense that the sum, that this purple vector plus this green vector is somehow going to be equal to this blue vector? I encourage you to pause the video and think about if that even makes sense. Well, one way to think about it is this first purple vector, it shifts us this much. It takes us from this point to that point. And so if we were to add it, let's start at this point and put the green vector's tail right there and see where it ends up putting us. So the green vector, we already have a version. So once again, we start the origin. Vector a takes us there. Now, let's start over there with the green vector and see where green vector takes us. And this makes sense. Vector a plus vector b. Put the tail of vector b at the head of vector a. So if you were to start at the origin, vector a takes you there then if you add on what vector b takes you, it takes you right over there. So relative to the origin, how much did you-- I guess you could say-- shift? And once again, vectors don't only apply to things like displacement. It can apply to velocity. It can apply to actual acceleration. It can apply to a whole series of things, but when you visualize it this way, you see that it does make complete sense. This blue vector, the sum of the two, is what results where you start with vector a. At that point right over there, vector a takes you there, then you take vector b's tail, start over there and it takes you to the tip of the sum. Now, one question you might be having is well, vector a plus vector b is this, but what is vector b plus vector a? Does this still work? Well, based on the definition we had where you add the corresponding components, you're still going to get the same sum vector. So it should come out the same. So this will just be negative 4 plus 6 is 2. 4 plus negative 2 is 2. But does that make visual sense? So if we start with vector b. So let's say you start right over here. Vector b takes you right over there. And then if you were to go there and you were to start with vector a-- so let's do that. So actually, let me make this a little bit-- actually, let me start with a new vector b. So let's say that that's our vector b right over there. And then-- actually, let me give this a place where I'll have some space to work with. So let's say that's my vector b right over there. And then let me get a copy of the vector a. That's a good one. So copy and let me paste it. So I could put vector a's tail at the tip of vector b, and then it'll take me right over there. So if I start right over here, vector b takes me there. And now I'm adding to that vector a, which starting here will take me there. And so from my original starting position, I have gone this far. Now, what is this vector? Well, this is exactly the vector 2, 2. Or another way of thinking about it, this vector shifts you 2 in the horizontal direction and 2 in the vertical direction. So either way, you're going to get the same result, and that should, hopefully, make visual or conceptual sense as well.