Hey, guys. So now that we've seen how to add vectors together, in some problems, you'll have to subtract them. In this video, I'm going to show you how to subtract vectors graphically. What we're going to see is that it's exactly like how we added vectors. You're going to combine a bunch of arrows, tip to tail. The only thing that's different here is that one or more of the vectors is going to get reversed. Let's check it out.
When we added vectors, just a quick recap, you would connect them tip to tail like this, a plus b, and the resultant vector was the shortest path from start to finish. The start was here, and the end of the last one is here, and this was my resultant vector. We called it c, and basically, we just can form a little triangle like this, and then we would count with the boxes to get the legs 34. And then to count to get the magnitude of the total displacement or the resultant, we just use the Pythagorean theorem for 3 squared and 4 squared, and that was 5. Now if we do b+a, we're going to do the same exact thing. The only thing that's different is that the vectors get reversed. Sorry. The vectors get added in reverse. So we just do b plus a, but the shortest path is still from start to finish like this, and then we end up with the same exact triangle because we're going to end with 34.
So what if instead of a+b, I want to do a minus b? Now, guys, the big difference here, the key difference is that when you're doing vector subtraction, one way you can think about this is we're just doing a plus the negative of b. So vector subtraction is really just vector addition. It's just that one of the vectors has a negative sign. So how does that work? Well, let's check it out.
So the a vector in our example was 2 to the right and 1 up. So if we wanted to add these things, we have to connect them tip to tail. So from the origin, I'm going to go 2 to the right and 1 up like this. So this is my a vector. And then if I want to add a and b, I'd have to go tip to tail, and my b vector pointed 1 to the right and 3 up. So I'm going to go 1 to the right and 3 up, and it would look like this. So this is my b vector over here. But I'm not trying to add a and b, I'm trying to add a and negative b. So what happens? What's the deal with that negative sign? Well, negative signs in physics just have to do with direction. So if positive b is 1 to the right and 3 up, then the negative of b is just going to be if I flip those two things. And I go one to the left and 3 down. So my negative b vector is going to point in this direction over here, basically exactly opposite. So notice how these two things have the same length, but they're pointing in perfectly opposite directions. So the negative of a vector is going to have the same magnitude, but it's going to point in the opposite direction.
So let's go ahead and add them now. So now we're not going to add these 2 vectors together. We're going to add these 2 vectors together. The resultant is still going to be the shortest path from start to finish. So the start is here, and the end of the last one's here so that my shortest path is just the straight line that connects those things. So, basically, it's if I'd actually just walked in this direction here. So we can make the little triangle. We can count out the boxes for the legs, 1 and 2, and the hypotenuse, either the magnitude, is going to be 1 squared plus 2 squared, and that's 2.24.
Now what if I wanted to do b minus a and basically reverse the order like I did before? Well, we can think about this using the same exact principle. B minus a is the same thing as b plus the negative of a. So now we're going to do the same exact thing, but we're just going to start off with the b vector first. My b vector points 1 to the right and 3 up, so it's going to look like this. And if I wanted to add b and a, I connect them tip to tail, and my a vector is 2 to the right and one up. So it looks like that. So this is my a vector. But this is positive a, so that means that negative a would just be pointing in the exact opposite direction. So it'd be 2 to the left and one down. So my negative a just points exactly in the opposite direction like this. And so I'm going to add these 2 vectors together. So this is my b, not this one, not this positive a. So my shortest path from start to finish is from the origin up to this point over here, the straight line. It's basically as if I had just walked in that direction instead of going here and then backwards like that. So this is my c. Break it up into a triangle. We're going to end up with 1 and 2. And so when we do the hypotenuse, we're going to use the same exact numbers when you get 1 squared and 2 squared, so the magnitude is going to be 2.24.
What is different though is the direction. So let me summarize. When we did vector addition, the order didn't matter. Whether we did a and b or b and a, we ended up at the same points. Our displacement vectors point in the same direction. When we do vector subtraction, however, the order does matter. You're really going to get the same magnitude, but here these vectors point exactly in opposite directions whether you do a minus b or b minus a. So be careful when you're adding those things, and make sure you're doing it in the proper order. Alright, guys. That's all there is to it. Let's go ahead and get to an example. So we're going to calculate the magnitude of this resultant vector here. We've got a minus b. So one way we can think about this is we're just doing a plus the negative of b. So let's check it out.
We've got these two vectors here, but they're actually not lined up tip to tail. They're both starting from the same place. So we've got our a vector like this. Now we have to add it tip to tail with a negative of b. So we're going to start over here. And now if b is going to be 2 to the right and 4 up, then my negative b is just going to be if I reverse it. So I'm going to go 2 to the left, and I'm going to go 1, 2, 3, 4 down. So my negative b vector looks like this, the exact opposite direction. So this is actually what I'm adding, a and negative b, which means that my resultant is just going to be the shortest path from start to finish as if I basically walked in this direction instead of doing here and that, instead of doing both of those motions there. So, this is my new displacement vector and my resultant. So I break it up into the triangle, count off the boxes. I've got 4 in this direction and 2 in this direction.
So the magnitude, using the Pythagorean theorem. So I've got 4 squared and 2 squared. You don't have to worry about, you know, the boxes pointing to the left or anything like that. So you just add the numbers and you're going to get 4.47. So let's call that meters. Alright, guys. That's all there is to it. Let me know if you have any questions.