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Adding & subtracting vectors end-to-end

Build intuition behind adding and subtracting vectors visually and the "end-to-end" method.

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Video transcript

- [Voiceover] Let's build our intuition for visually adding and subtracting vectors. So let's say that I have vector a, and I add that to vector, to vector b, and the resulting vector is a vector c, is vector c. So what could this look like visually if we assume that a, b and c are two-dimensional vectors? Well, I'm just going to draw what vector a might look like. So let's say that this right over here is vector a, and vector b, vector b, since I'm adding it to vector a, I'm going to put its initial point at the terminal point of vector a, and then I'm going to draw vector b. So let's say vector b looks like that. So that is vector b, let me label these. That is vector a, this is vector b. And I did that so that I can figure out what the sum is, what vector c is going to be. So that is vector b, and what would that be? Well, we would start at the initial point of vector a and then go to the terminal point of vector b. So this right over there would be the sum. So that would be vector c right over there. So the important realization is if I add two vectors, I would put the tail of one at the head of the other. And now what's neat about this if I'm adding, the order doesn't matter. I could've done this the other way around. I could've started with vector b. I could've said vector b plus vector a is equal to vector c, is equal to vector c, and you could see that visually. It would be a slightly different visual diagram but you get to the same place. So if I start with vector b, let's say I start over here, In fact, you don't have to start at the origin but let's say that was the origin. So I could start with vector b, draw vector b just like that, and then add vector a to it. So start vector a at the terminal point of vector b, and then go to (mumbles) just draw vector a. So vector a. So once again, a vector, I can shift them around as long as I'm not changing the direction or their magnitude. So vector a looks like that. And notice, if you now go and start at the initial point of b and go to the terminal point of a, you still get vector c. So that's why a plus b and b plus a are going to give you the same thing. Now what if instead of saying a plus b I wanted to think about what a minus b is going to be? So let me write that down. Vector a minus vector b, minus vector b. And let's call that vector d. That is equal to vector d. So once again, I could start with vector a and here, order matters. So vector a looks something like this. this is hand-drawn so it's not going to be completely perfect. So vector a, just like that. And one way of thinking about subtracting vector b is instead of adding vector b the way we did here, we could add negative b. So negative b would have the same magnitude but just the opposite direction. So that's vector a. Vector negative b will still start right over here, but will go in the opposite direction. So let's do that. So negative b is going to look like this, is going to look something like this. So that is negative b. Notice, same magnitude exactly opposite direction. We've flipped it around 180 degrees, and now the resulting vector is going to be d. So vector d is going to look like that, vector d. So c is a plus b, d is a minus b. Or you can even call this a plus, a plus negative b. Now with that out of the way, let's draw some diagrams and go the other way. See if we could go from the diagrams to the actual equations. So let's start with, let me draw an interesting one. So let's say that this... Let's say that's vector a. Vector a. I'll use green. Let's say that, that is vector b, and I will now use magenta. And let's say that this is vector c. Vector c. So I encourage you to pause the video and see if you can write an equation that defines this relationship. Well, this is interesting because they're all going in a circle right over here. Let's say that you started at... This is your initial point. You said, okay, a plus b, well, if you're trying to figure out what a plus b is going to be, the resulting vector would start here and end here. But vector c is going in the opposite direction. But we could, instead of thinking about vector c like this, we could think about the opposite of vector c, which would do, so instead of calling this c, I could flip this around by calling it this negative c. So I could flip this around, and now, let me do the same color, this would be equal to negative c. Notice, before I just had vector c here and it started at this point and ended at this point. Now I just flipped it around, it has the exact opposite direction, same magnitude. Now it is negative c. And this makes it easier for us to construct an equation because negative c starts at the tip, at the initial point or the tail of vector a, and it goes to the head of vector b or the terminal point of vector b. So we can now write an equation. We could say vector a plus vector b, plus vector b, is equal to, is equal to not c, it's equal to the negative of vector c. So hopefully, you found that interesting.