3. Vectors

Adding Vectors Graphically

# Subtracting Vectors Graphically

Patrick Ford

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Hey, guys. So now that we've seen how to add vectors together and some problems, you'll have to subtract them. So in this video, I'm gonna 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 gonna combine a bunch of arrows tipped the 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. So 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 ones here, and this was my result in Vector. We called it C, and basically we just conform a little triangle like this. And then we would counter the boxes to get the legs three and four on that account to get the magnitude of the total displacement or the resultant. We just use the Pythagorean theory for three squared four squared and that was five. Now, if we do B plus A, we're gonna do the same exact thing. The only thing that's different is that the vectors get reversed. Uh, sorry that 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 gonna end up with three and four. So what if instead of a plus B, I want to do a minus bi 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 be 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 one or to the right and one up. So if you want to add these things, we have to connect them tip to tail. So from the origin, I'm gonna go to the right and one up like this. So this is my a vector. And then if I want to add a and B, I would have to go tip to tail and might be vector pointed one to the right and three up. So I'm gonna go one to the right and three 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 negative. Be. 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 one to the right and three up, then the negative of B is just gonna be If I flip those two things and I go one to the left and three down, So by negative be vector is gonna 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 signs are perfectly opposite directions. So the negative of a vector is gonna have the same magnitude, but it's gonna points in the opposite direction. So let's go ahead and add them now. So now we're not gonna add these two vectors together. We're gonna add these two vectors together. The result is still gonna be the shortest path from start to finish the starts here and the end of last ones here. So that my shortest path, it's just the straight line that connects those things. So basically, it's if I'd actually just walked in this direction here so we couldn't make the little triangle weaken Count of the boxes for the legs one and two and the high pot news or the magnitude is gonna be one squared plus two 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 we're one week we can think about. This is B plus the negative of a So now we're gonna do the same exact thing. We're just gonna start off with the vector First, my be vector points one to the right and three up. So it's gonna look like this. And if I if I wanted to add being a, I connect them tip to tail and my A vector is to the right and one up. So it looks like that. So this is a vector, but this is positive. A So that means that negative A would just be pointing in the exact opposite direction. So would be to the left and one down. So my negative A just points exactly the opposite direction. Like this. And so I'm gonna add these two 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 is 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 gonna end up with one and two. And so when we do the high pot news, we're gonna use the same exact numbers when you get one squared in two squared, so the magnitude is going to be 2.24 What is different, though, is the direction. So let me summarize. We did Vector edition. The order didn't matter whether we did A and B or B and a we ended up at the same points are displacement vectors point in the same direction we do vector subtraction. However, the order does matter, you're gonna get the same magnitude. But here these vectors point exactly in opposite directions whether you do a minus bi or B minus a. So be careful when you're adding those things on, make sure you're doing 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 gonna calculate the magnitude of this result in vector here. We've got a minus bi. So one way we can think about this is we're just doing a plus the negative of be 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 they're both basically starting from the same place. So we've got our A vector like this and now we have to add it tip to tail with negative of be So we're gonna start over here and now if b is gonna be to the right and for up then my negative B is just gonna be if I reverse it. So I'm gonna go to to the left and I'm gonna go 1234 down to my negative be vector. Looks like this the exact opposite direction. So this is actually what I'm adding A a negative B. Which means that my resultant is just gonna be the shortest path from start to finish is 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. My resultant. And so it break it up into the triangle kind of the boxes. I've got four in this direction and to in this direction. So the magnitude, which is what I'm looking for here, it's just gonna be the Pythagorean theorem. So I've got four squared and two square Don't have to worry about, you know, the boxes points in the left or anything like that. So you just add up the numbers and you're gonna get four points 47. So let's call that meters. Alright, guys, that's all there is to it. Let me know if you have any questions

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